Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems of Integral Equations

*Massoumeh Poura'bd Rokn Saraei and Mashaallah Matinfar*

### **Abstract**

This work aims at focusing on modifying the moving least squares (MMLS) methods for solving two-dimensional linear and nonlinear systems of integral equations and system of differential equations. The modified shape function is our main aim, so for computing the shape function based on the moving least squares method (MLS), an efficient algorithm is presented. In this modification, additional terms is proposed to impose based on the coefficients of the polynomial base functions on the quadratic base functions (m = 2). So, the MMLS method is developed for solving the systems of two-dimensional linear and nonlinear integral equations at irregularly distributed nodes. This approach prevents the singular moment matrix in the context of MLS based on meshfree methods. Also, determining the best radius of the support domain of a field node is an open problem for MLS-based methods. Therefore, the next important thing is that the MMLS algorithm can automatically find the best neighborhood radius for each node. Then, numerical examples are presented that determine the main motivators for doing this so. These examples enable us to make comparisons of two methods: MMLS and classical MLS.

**Keywords:** moving least squares, modified moving least squares, systems of integral equations, algorithm of shape function, numerical solutions

**MSC 2010:** 45G15, 45F05,45F35, 65D15

### **1. Introduction**

In mathematics, there are many functional equations of the description of a real system in the natural sciences (such as physics, biology, Earth science, meteorology) and disciplines of engineering. For instance, we can point to some mathematical model from physics that describe heat as a partial differential equation and the inverse problem of it's as integro-differential equations. Also, another example in nature is Laplace's equation which corresponds to the construction of potential for a vector field whose effect is known at the boundary of Domain alone. Especially, the integral equations have wide applicability which has been cited in [1–4].

However, there are many significant analytical methods for solving integral equations but most of them especially in nonlinear cases, finding an analytical representation of the solution is so difficult, therefore, it is required to obtain approximate solutions. The interested reader can find several numerical methods for approximating the solution of these problems in [5–14] and the references therein.

function is a set of nodes in the problem domain that just those points directly contributes to the construction of the shape function, so the MLS shape function is locally supported. According to the classical least squares method, an optimization

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

*uh* **x***<sup>j</sup>*

Therefore, in the MLS approach, a weight optimization problem will be solved which is dependent on nodal points. We start, with the basic idea of taking a set of the nodal points in Ω so that Ω ⊆ *<sup>d</sup>:* Also Ω**<sup>x</sup>** ⊆ Ω is neighboring nodes of point **x** and finding an approximation function with *m* basis functions, in a system with *n*

� � � *<sup>u</sup>* **<sup>x</sup>***<sup>j</sup>* � � � � <sup>2</sup> !

Where Ideally the approximation function *uh*ð Þ *<sup>x</sup>* should match the function *<sup>u</sup>*ð Þ **<sup>x</sup>** .

*T U*ð Þ¼ *F*

where *T* consists of linear and nonlinear operators and *U* ¼ ð Þ *u*1, *u*2, … , *un* is

So for the approximation of any of the *ui*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* in <sup>Ω</sup>**x**, <sup>∀</sup>*x*<sup>∈</sup> <sup>Ω</sup>**x**, *<sup>u</sup><sup>h</sup>*

*aj*ð Þ **x** *pj*

is a vector containing unknown coefficients *aj*ð Þ **x** , *j* ¼ 1, 2, … *m* dependent on the intrest point position. Also *m* unknown functions **a x**ð Þ¼ ð Þ *a*1ð Þ **x** , *a*2ð Þ *x* , … *am*ð Þ **x**

is minimized. Note that the weight function *wi*ð Þ **x** is associated with node *j*. As we know, each redial basis function that define in [31] can be used as weight

between **<sup>x</sup>** and **<sup>x</sup>***j*<sup>Þ</sup> and *<sup>ϕ</sup>* : *<sup>d</sup>* ! is a nonnegative function with compact support. In this chapter, we will use following weight functions and will compare them to

<sup>1</sup> � exp �*δ*<sup>2</sup>

� exp �*δ*<sup>2</sup>

*c*2

0 *elsewhere:*

*c*2 � �

� � <sup>0</sup>≤*r*≤*<sup>δ</sup>*

*δ*

each other, corresponding to the node *j*, in the numerical examples.

exp �*r*<sup>2</sup> *c*2 � �

� � a set of polynomial of degree at most *<sup>m</sup>*, *<sup>m</sup>* <sup>∈</sup> *:* Let **a x**ð Þ

� � is the known vector of

ð Þ¼ **<sup>x</sup>** *PT*ð Þ **<sup>x</sup>** *<sup>a</sup>*ð Þ **<sup>x</sup>** *:* (1)

*wi*ð Þ¼ **<sup>x</sup>** *<sup>P</sup>:***<sup>a</sup>** � **<sup>u</sup>***<sup>i</sup>* ½ �*<sup>T</sup>:W: <sup>P</sup>:***<sup>a</sup>** � **<sup>u</sup>***<sup>i</sup>* ½ �, (2)

� � where *<sup>r</sup>* <sup>¼</sup> k k **<sup>x</sup>** � **<sup>x</sup>***<sup>i</sup>* <sup>2</sup> (the Euclidean distance

*<sup>i</sup>* ð Þ **x** can

(3)

*min* <sup>X</sup>*<sup>m</sup>*

the unknown vector of functions, also *F* ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>n</sup>*

*uh*

*<sup>i</sup>* ð Þ¼ **<sup>x</sup>** <sup>X</sup>*<sup>m</sup>*

� �**a x**ð Þ� *ui* **<sup>x</sup>***<sup>j</sup>* � � � � <sup>2</sup>

*j*¼1

*j*¼1

problem should be solved as follows

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

equations as

functions.

be defined as

Let **P** ¼ *p*1, *p*2, … *pm*

are chosen such that:

*<sup>J</sup>*ð Þ¼ **<sup>x</sup>** <sup>X</sup>*<sup>m</sup>*

*j*¼1

function, we can define *wj*ð Þ¼ *<sup>r</sup> <sup>ϕ</sup> <sup>r</sup>*

a. Guass weight function

b. RBF weight function

**59**

*w r*ð Þ¼

8 >>>>><

>>>>>:

**P***<sup>T</sup>* **x***<sup>j</sup>*

Moreover, there are various numerical and analytical methods have been used to estimate the solution of integrodifferential equations or Abels integral equations [12, 15–18]. Recently the meshless based methods, particularly Moving Least Squares (MLS) method, for a solution of partial differential equations and ordinary differential equations have been paid attention. Using this approach some new methods such as meshless local boundary integral equation method [19], Boundary Node Method (BNM) [20], moving least square reproducing polynomial meshless method [21] and other relative methods are constructed. The new class of meshless methods has been developed which only relied on a set of nodes without the need for an additional mesh in the solution of a one-dimensional system of integral equations [22].

A local approximation of unknown function presented in the MLS method give us to possible choose the compact support domain for each data point as a sphere or a parallelogram box centered on a point [23, 24]. So each data point has an associated with the size of its compact support domain as dilatation parameter. Therefore the number of data point and dilatation parameter are direct effects on the MLS, Also by increasing the degree of the polynomial base function for complex data distributions give a more validated fashion. Nevertheless, in this case, it becomes more difficult to ensure the independence of the shape functions, and the leastsquares minimization problem becomes ill-posed.

The common solution for increased the number of admissible node distribution is increasing the size of the support domains (a valid node distribution is referred to as an œadmissible node distribution [23]). There have been several proposed for choosing the radius of support domain [25], but one of the efficient suggestion was raised by Chen shen [26]. The author in [27] has introduced a new algorithm for selecting the suitable radius of the domain of influence. Also in [28], presented a modified MLS(MMLS) approximation on the shape function generation algorithm with additional terms based on the coefficients of the polynomial basis functions. It is an efficient method which has been proposed for handling a singular moment matrix in the MLS based methods. The advantage of this method compared to methods based on mesh such as a finite element or finite volume is this the domain of the problem is not important because this approximation method is based on a set of scattered points instead of domain elements for interpolation or approximation. So the geometry of the domain does not interfere in the MLS.

#### **2. Methodology**

#### **2.1 Introduction of the MLS approximation**

The Moving Least Square (MLS) method is a feasible numerical approximation method that is an extension of the least squares method, also it is the component of the class of meshless schemes that have a highly accurate approximation. The MlS approximation method is a popular method used in the many meshless methods [12, 19, 21, 22, 29, 30]. In many procedures used to construct the MLS shape function is used support-domain concept. The support domain of the shape

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

function is a set of nodes in the problem domain that just those points directly contributes to the construction of the shape function, so the MLS shape function is locally supported. According to the classical least squares method, an optimization problem should be solved as follows

$$\min \left( \sum\_{j=1}^{m} \left( u^h(\mathbf{x}\_j) - u\left(\mathbf{x}\_j\right) \right)^2 \right)$$

Where Ideally the approximation function *uh*ð Þ *<sup>x</sup>* should match the function *<sup>u</sup>*ð Þ **<sup>x</sup>** . Therefore, in the MLS approach, a weight optimization problem will be solved which is dependent on nodal points. We start, with the basic idea of taking a set of the nodal points in Ω so that Ω ⊆ *<sup>d</sup>:* Also Ω**<sup>x</sup>** ⊆ Ω is neighboring nodes of point **x** and finding an approximation function with *m* basis functions, in a system with *n* equations as

$$T(U) = F$$

where *T* consists of linear and nonlinear operators and *U* ¼ ð Þ *u*1, *u*2, … , *un* is the unknown vector of functions, also *F* ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>n</sup>* � � is the known vector of functions.

So for the approximation of any of the *ui*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* in <sup>Ω</sup>**x**, <sup>∀</sup>*x*<sup>∈</sup> <sup>Ω</sup>**x**, *<sup>u</sup><sup>h</sup> <sup>i</sup>* ð Þ **x** can be defined as

$$u\_i^h(\mathbf{x}) = \sum\_{j=1}^m a\_j(\mathbf{x}) p\_j(\mathbf{x}) = P^T(\mathbf{x}) a(\mathbf{x}).\tag{1}$$

Let **P** ¼ *p*1, *p*2, … *pm* � � a set of polynomial of degree at most *<sup>m</sup>*, *<sup>m</sup>* <sup>∈</sup> *:* Let **a x**ð Þ is a vector containing unknown coefficients *aj*ð Þ **x** , *j* ¼ 1, 2, … *m* dependent on the intrest point position. Also *m* unknown functions **a x**ð Þ¼ ð Þ *a*1ð Þ **x** , *a*2ð Þ *x* , … *am*ð Þ **x** are chosen such that:

$$J(\mathbf{x}) = \sum\_{j=1}^{m} \left( \mathbf{P}^T(\mathbf{x}\_j) \mathbf{a}(\mathbf{x}) - u\_i(\mathbf{x}\_j) \right)^2\\w\_i(\mathbf{x}) = \left[ P.\mathbf{a} - \mathbf{u}\_i \right]^T.W.[P.\mathbf{a} - \mathbf{u}\_i],\tag{2}$$

is minimized. Note that the weight function *wi*ð Þ **x** is associated with node *j*. As we know, each redial basis function that define in [31] can be used as weight function, we can define *wj*ð Þ¼ *<sup>r</sup> <sup>ϕ</sup> <sup>r</sup> δ* � � where *<sup>r</sup>* <sup>¼</sup> k k **<sup>x</sup>** � **<sup>x</sup>***<sup>i</sup>* <sup>2</sup> (the Euclidean distance between **<sup>x</sup>** and **<sup>x</sup>***j*<sup>Þ</sup> and *<sup>ϕ</sup>* : *<sup>d</sup>* ! is a nonnegative function with compact support. In this chapter, we will use following weight functions and will compare them to each other, corresponding to the node *j*, in the numerical examples.

a. Guass weight function

$$w(r) = \begin{cases} \exp\left(\frac{-r^2}{c^2}\right) - \exp\left(\frac{-\delta^2}{c^2}\right) \\ \hline \mathbf{1} - \exp\left(\frac{-\delta^2}{c^2}\right) \\ \mathbf{0} \end{cases} \quad \mathbf{0} \le r \le \delta \tag{3}$$

b. RBF weight function

However, there are many significant analytical methods for solving integral equations but most of them especially in nonlinear cases, finding an analytical representation of the solution is so difficult, therefore, it is required to obtain approximate solutions. The interested reader can find several numerical methods for approximating the solution of these problems in [5–14] and the references

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Moreover, there are various numerical and analytical methods have been used to estimate the solution of integrodifferential equations or Abels integral equations [12, 15–18]. Recently the meshless based methods, particularly Moving Least Squares (MLS) method, for a solution of partial differential equations and ordinary differential equations have been paid attention. Using this approach some new methods such as meshless local boundary integral equation method [19], Boundary Node Method (BNM) [20], moving least square reproducing polynomial meshless method [21] and other relative methods are constructed. The new class of meshless methods has been developed which only relied on a set of nodes without the need for an additional mesh in the solution of a one-dimensional system of integral

A local approximation of unknown function presented in the MLS method give us to possible choose the compact support domain for each data point as a sphere or a parallelogram box centered on a point [23, 24]. So each data point has an associated with the size of its compact support domain as dilatation parameter. Therefore the number of data point and dilatation parameter are direct effects on the MLS, Also by increasing the degree of the polynomial base function for complex data distributions give a more validated fashion. Nevertheless, in this case, it becomes more difficult to ensure the independence of the shape functions, and the least-

The common solution for increased the number of admissible node distribution is increasing the size of the support domains (a valid node distribution is referred to as an œadmissible node distribution [23]). There have been several proposed for choosing the radius of support domain [25], but one of the efficient suggestion was raised by Chen shen [26]. The author in [27] has introduced a new algorithm for selecting the suitable radius of the domain of influence. Also in [28], presented a modified MLS(MMLS) approximation on the shape function generation algorithm with additional terms based on the coefficients of the polynomial basis functions. It is an efficient method which has been proposed for handling a singular moment matrix in the MLS based methods. The advantage of this method compared to methods based on mesh such as a finite element or finite volume is this the domain of the problem is not important because this approximation method is based on a set of scattered points instead of domain elements for interpolation or approxima-

The Moving Least Square (MLS) method is a feasible numerical approximation method that is an extension of the least squares method, also it is the component of the class of meshless schemes that have a highly accurate approximation. The MlS approximation method is a popular method used in the many meshless methods [12, 19, 21, 22, 29, 30]. In many procedures used to construct the MLS shape function is used support-domain concept. The support domain of the shape

tion. So the geometry of the domain does not interfere in the MLS.

therein.

equations [22].

**2. Methodology**

**58**

squares minimization problem becomes ill-posed.

**2.1 Introduction of the MLS approximation**

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$w(r) = \begin{cases} (1-r)^6(6+36r+82r^2+72r^3+30r^4+5r^5) & 0 \le r \le \delta \\ 0 & \text{elsewhere} \end{cases} \tag{4}$$

c. Spline weight function

$$w(r) = \begin{cases} 1 - 6\left(\frac{r}{\delta}\right)^2 + 8\left(\frac{r}{\delta}\right)^3 - 3\left(\frac{r}{\delta}\right)^4 & 0 \le r \le \delta\\ 0 & \text{elsewhere} \end{cases} \tag{5}$$

*D<sup>α</sup>ui***x** ≈ *D<sup>α</sup>uh*

**2.2 Modify algorithm of MLS shape function**

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

size of compact support domain in a node point **x***i*.

ℵ *j*j*xj* ∈ Ω*δ<sup>i</sup>*

3: *α* 0*:*01 (This value selected experimentally.)

localization at a fixed point *x*)

9: Compute *wi* for any *xj* ∈ Ω*<sup>i</sup>*

*<sup>ε</sup>* **then**

*i*

that for any point *xi* ∈ Ω, *i* ¼ 1, 2, … , *N*, [31].

matrix *A* is nonsingular.

the algorithm.

**Algorithm 1**

function to be evaluated. 1: **procedure** MATRIX A

> ∥*x* � *xi*∥2) 5: Loop

7: **for** *j*∈*I x*ð Þ **do** 8: **for** *i* ¼ 1 : *N* **do**

13: **if** *cond A*ð Þ<sup>≥</sup> <sup>1</sup>

15: *λnew* ¼ *λnew* þ *αλ* 16: *δ* ¼ *λnew* � *h*

20: **if** *δ<sup>i</sup>* ≤ ∥*X*Ω∥<sup>2</sup> **then** 21: **goto** *Loop*

11: **end** 12: **end**

14: {

17: **else** 18: **goto** *end*

19: }

22: **end**

**61**

10: *<sup>A</sup>* <sup>¼</sup> *<sup>A</sup>* <sup>þ</sup> *wip*<sup>2</sup>

2: *λnew λ*

*<sup>i</sup>* ð Þ¼ **<sup>x</sup>** <sup>X</sup> *N*

*j*¼1

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

In the MLS approximation method, a local evaluation of the approximating unknown function is desired, and therefore for any nodal points the compact support domain is chosen as a sphere or a parallelogram box centered on the point [23, 29, 32]. This finding which the support domains contain what points. Each data point has a connected dilatation parameter *λ* which is given *δ<sup>i</sup>* ¼ *λhi*. Also, *δ<sup>i</sup>* is the

Also, the necessary condition for that the moment matrix A be nonsingular is

� � � � <sup>≥</sup> *<sup>m</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>*

In the remainder of this section, we introduce the new algorithm, with the aim of avoiding the singularity of the matrix A by choosing the correct *λ* parameter by

**Require:** *X* ¼ *xi* f g : *i* ¼ 1, 2, … , *N* - Coordinates of points whose MLS shape

4: *δ* ¼ *λnew* � *h* (h: the fill distance is defined to be *h* ¼ sup*<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> min <sup>1</sup>≤*<sup>j</sup>* <sup>≤</sup> *<sup>N</sup>*

6: set *I x*ð Þ¼ f g *j*∈f g 1, 2, … , *N* , ∥*x* � *xi*∥<sup>2</sup> ≤*δ* (Using set of indices *I*, by

So the dilatation parameters *λ* determine the number of points of support domain, Also these points participate in the production of the shape function Therefore, *λ* is very important and should be chosencorrectly so that the moment

*<sup>D</sup><sup>α</sup>aj*ð Þ **<sup>x</sup>** *ui* **<sup>x</sup>***<sup>j</sup>*

� �, *x*∈ Ω (14)

Where *c* is constant and is called shape parameter. Also *δ* is the size of support domain.

*N* is the number of nodes in Ω**<sup>x</sup>** with *wi*ð Þ *x* >0, the matrices *P* and *W* are defined as

$$P = \begin{bmatrix} \mathbf{p}^T(\mathbf{x}\_1), \mathbf{p}^T(\mathbf{x}\_2), \dots \mathbf{p}^T(\mathbf{x}\_N) \end{bmatrix}\_{N \times (m+1)}^T \tag{6}$$

$$\mathcal{W} = \text{diag}((w\_i(\mathbf{x})), i = \mathbf{1}, \mathbf{2}, \dots, N\tag{7}$$

and

$$\mathbf{u}^{h} = \begin{bmatrix} u\_1^h, u\_2^h, \dots, u\_n^h \end{bmatrix}. \tag{8}$$

It is important to note that *uT <sup>i</sup>* , *i* ¼ 1, 2, … *n*, in (2) and (8) are the artificial nodal values, and not the nodal values of the unknown trial function *<sup>u</sup><sup>h</sup>*ð Þ **<sup>x</sup>** in general. With respect to **a x**ð Þ and *uT <sup>i</sup>* will be obtained,

$$A(\mathbf{x})a(\mathbf{x}) = B(\mathbf{x})\mathbf{u}\_{\text{i}},\tag{9}$$

where the matrices *A*ð Þ **x** and *B*ð Þ **x** are defined by:

$$B(\mathbf{x}) = [w\_1 \mathbf{p}(\mathbf{x}\_1), w\_2 \mathbf{p}(\mathbf{x}\_2), \dots, w\_N \mathbf{p}(\mathbf{x}\_N)] \tag{10}$$

$$A(\mathbf{x}) = \sum\_{i=1}^{N} w\_i(\mathbf{x}) \mathbf{p}^T(\mathbf{x}\_i) \mathbf{p}(\mathbf{x}\_i) = \mathbf{p}^T(\mathbf{x}) w(\mathbf{x}) \mathbf{p}(\mathbf{x}).\tag{11}$$

The matrix *A*ð Þ **x** in (11) is non-singular when the rank of matrix *P*ð Þ **x** equals to *m* and vice versa. In such a case, the MLS approximation is well-defined. With computing **a x**ð Þ, *uh <sup>i</sup>* can be obtained as follows:

$$u\_i^h(\mathbf{x}) = \sum\_{j=1}^N \phi\_j(\mathbf{x}) u\_i(\mathbf{x}\_j) = \boldsymbol{\rho}^T.\mathbf{u}\_i\tag{12}$$

*ϕj*ð Þ **x** is called the shape function of the MLS approximation corresponding to the nodal point **x***j*, where

$$\boldsymbol{\rho}(\mathbf{x}) = \mathbf{p}^T(\mathbf{x}) \boldsymbol{A}^{-1}(\mathbf{x}) \boldsymbol{B}(\mathbf{x}) \tag{13}$$

Also with use the weight function, matrix *A*, *B* are computed and then *ϕi*ð Þ *x* is determined from (13), If, further, *ϕ* is sufficiently smooth, derivatives of *U* are usually approximated by derivatives of *U<sup>h</sup>*,

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

$$D^a u\_i \mathbf{x} \approx D^a u\_i^h(\mathbf{x}) = \sum\_{j=1}^N D^a a\_j(\mathbf{x}) u\_i(\mathbf{x}\_j), \mathbf{x} \in \Omega \tag{14}$$

#### **2.2 Modify algorithm of MLS shape function**

In the MLS approximation method, a local evaluation of the approximating unknown function is desired, and therefore for any nodal points the compact support domain is chosen as a sphere or a parallelogram box centered on the point [23, 29, 32]. This finding which the support domains contain what points. Each data point has a connected dilatation parameter *λ* which is given *δ<sup>i</sup>* ¼ *λhi*. Also, *δ<sup>i</sup>* is the size of compact support domain in a node point **x***i*.

Also, the necessary condition for that the moment matrix A be nonsingular is that for any point *xi* ∈ Ω, *i* ¼ 1, 2, … , *N*, [31].

$$\aleph(\{j|\mathbf{x}\_j \in \Omega\_{\mathbb{N}}\}) \ge m, \ j = 1, 2, \dots, N$$

So the dilatation parameters *λ* determine the number of points of support domain, Also these points participate in the production of the shape function Therefore, *λ* is very important and should be chosencorrectly so that the moment matrix *A* is nonsingular.

In the remainder of this section, we introduce the new algorithm, with the aim of avoiding the singularity of the matrix A by choosing the correct *λ* parameter by the algorithm.

#### **Algorithm 1**

*w r*ð Þ¼ ð Þ <sup>1</sup> � *<sup>r</sup>*

c. Spline weight function

It is important to note that *uT*

With respect to **a x**ð Þ and *uT*

domain.

defined as

and

puting **a x**ð Þ, *uh*

**60**

the nodal point **x***j*, where

*w r*ð Þ¼ <sup>1</sup> � <sup>6</sup> *<sup>r</sup>*

(

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*δ* � �<sup>2</sup> <sup>þ</sup> <sup>8</sup> *<sup>r</sup> δ* � �<sup>3</sup> � <sup>3</sup> *<sup>r</sup> δ*

(

<sup>6</sup> <sup>6</sup> <sup>þ</sup> <sup>36</sup>*<sup>r</sup>* <sup>þ</sup> <sup>82</sup>*r*<sup>2</sup> <sup>þ</sup> <sup>72</sup>*r*<sup>3</sup> <sup>þ</sup> <sup>30</sup>*r*<sup>4</sup> <sup>þ</sup> <sup>5</sup>*r*<sup>5</sup> ð Þ <sup>0</sup><sup>≤</sup> *<sup>r</sup>*<sup>≤</sup> *<sup>δ</sup>*

0 *elsewhere:*

Where *c* is constant and is called shape parameter. Also *δ* is the size of support

*N* is the number of nodes in Ω**<sup>x</sup>** with *wi*ð Þ *x* >0, the matrices *P* and *W* are

*<sup>P</sup>* <sup>¼</sup> **<sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>**<sup>1</sup> , **<sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>**<sup>2</sup> , … **<sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>***<sup>N</sup>* � �*<sup>T</sup>*

**<sup>u</sup>***<sup>h</sup>* <sup>¼</sup> *uh*

*<sup>i</sup>* will be obtained,

where the matrices *A*ð Þ **x** and *B*ð Þ **x** are defined by:

*i*¼1

*<sup>i</sup>* can be obtained as follows:

*uh*

usually approximated by derivatives of *U<sup>h</sup>*,

*<sup>i</sup>* ð Þ¼ **<sup>x</sup>** <sup>X</sup> *N*

*j*¼1 *ϕj*

*<sup>φ</sup>*ð Þ¼ **<sup>x</sup> <sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>** *<sup>A</sup>*�<sup>1</sup>

*<sup>A</sup>*ð Þ¼ **<sup>x</sup>** <sup>X</sup> *N*

<sup>1</sup> , *uh* 2, … *uh n*

The matrix *A*ð Þ **x** in (11) is non-singular when the rank of matrix *P*ð Þ **x** equals to *m* and vice versa. In such a case, the MLS approximation is well-defined. With com-

ð Þ **x** *ui* **x***<sup>j</sup>*

*ϕj*ð Þ **x** is called the shape function of the MLS approximation corresponding to

Also with use the weight function, matrix *A*, *B* are computed and then *ϕi*ð Þ *x* is determined from (13), If, further, *ϕ* is sufficiently smooth, derivatives of *U* are

values, and not the nodal values of the unknown trial function *<sup>u</sup><sup>h</sup>*ð Þ **<sup>x</sup>** in general.

� �<sup>4</sup> 0≤ *r*≤*δ*

*W* ¼ *diag w*ðð Þ *<sup>i</sup>*ð Þ **x** , *i* ¼ 1, 2, … , *N* (7)

� �*:* (8)

*<sup>i</sup>* , *i* ¼ 1, 2, … *n*, in (2) and (8) are the artificial nodal

*A*ð Þ **x** *a*ð Þ¼ **x** *B*ð Þ **x u***i*, (9)

*wi*ð Þ **<sup>x</sup> <sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>***<sup>i</sup>* **p x**ð Þ¼*<sup>i</sup>* **<sup>p</sup>***<sup>T</sup>*ð Þ **<sup>x</sup>** *<sup>w</sup>*ð Þ **<sup>x</sup> p x**ð Þ*:* (11)

� � <sup>¼</sup> *<sup>φ</sup><sup>T</sup>:***u***<sup>i</sup>* (12)

ð Þ **x** *B*ð Þ **x** (13)

*B*ð Þ¼ **x** ½ � *w*1**p x**ð Þ<sup>1</sup> , *w*2**p x**ð Þ<sup>2</sup> , … , *wN***p x**ð Þ *<sup>N</sup>* (10)

*<sup>N</sup>*�ð Þ *<sup>m</sup>*þ<sup>1</sup> (6)

(4)

(5)

0 *elsewhere:*

**Require:** *X* ¼ *xi* f g : *i* ¼ 1, 2, … , *N* - Coordinates of points whose MLS shape function to be evaluated.

1: **procedure** MATRIX A


*i*

```
10: A ¼ A þ wip2
```

```
11: end
```
12: **end**

```
13: if cond Að Þ≥ 1
                   ε then
```

In Algorithm 1.

*X*: is a set containing N scattered points which are called centers or data site and I (x) is the Index of points which MLS shape function is evaluated.

where, *Oi*,*<sup>j</sup>* is the null matrix. By minimizing the functional (18), the coefficients

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

And the matrics *B*ð Þ **x** is determined as the same before. So we have

*<sup>φ</sup>m*ð Þ¼ **<sup>x</sup> a x**ð Þ¼ *pT*ð Þ **<sup>x</sup>** *<sup>A</sup>*�<sup>1</sup>

**3. Stiff systems of ordinary differential equations**

*B U*,

*uh*

*ai*ð Þ **x** *u*2ð Þ **x***<sup>i</sup>* , … ,

*ai*ð Þ **x** *u*2ð Þ **x***<sup>i</sup>* , … ,

!

!

*<sup>j</sup>* ð Þ¼ **<sup>x</sup>** <sup>X</sup> *N*

> X*N i*¼1

> > X*N i*¼1

*i*¼1

where *j* ¼ 1, 2, … , *n* is the number of unknown functions. we estimate the unknown functions *ui* by (25), so the system (24) becomes the following form

*ai*ð Þ **x** *un*ð Þ **x***<sup>i</sup>*

*ai*ð Þ **x** *un*ð Þ **x***<sup>i</sup>*

þ

*∂U ∂***x** � �

*F*ð Þ **x** is a vector of known analytical functions on the domain Ω and ∂Ω is the boundary of Ω. The operator can be divided by *A* ¼ *L* þ *N*, where *L* is the linear part, and *N* is the nonlinear part of its. So (23) can be, rewritten as follows;

where *φm*ð Þ **x** is the shape function of the MMLS approximation method.

In this section, we use MLS approximation method for numerical solution of the Stiff system of ordinary differential equations so consider the following differential

*A U*ð Þ� *F*ð Þ¼ **x** 0, *U*ð Þ¼ 0 *U*0, **x**∈ Ω (23)

*L U*ð Þþ *N U*ð Þ� *F*ð Þ¼ **x** 0 (24)

*<sup>j</sup>* from (13). So we have

*ai*ð Þ **x** *uj*ð Þ **x***<sup>i</sup>* (25)

<sup>¼</sup> *<sup>f</sup>* <sup>1</sup>ð Þ **<sup>x</sup>** , *<sup>f</sup>* <sup>2</sup>ð Þ **<sup>x</sup>** , … , *<sup>f</sup> <sup>n</sup>*ð Þ **<sup>x</sup>** � � <sup>þ</sup> **r x**ð Þ*:*

(26)

¼ 0, **x**∈ ∂Ω*:*

where A is a general differential operator, *U*<sup>0</sup> is an initial approximation of (23),

We assume that **a** ¼ f g *a*1, *a*2, … , *am* are the MLS shape functions so in order to solve system (24), *N* nodal points *xi* are selected on the Ω, which *xi* f g j*i* ¼ 1, 2, … , *N* is q-unisolvent. The distribution of nodes could be selected regularly or randomly.

*A*ð Þ **x a**ð Þ¼ *x B*ð Þ **x u***i*, (20)

*<sup>A</sup>* <sup>¼</sup> *PT:W:<sup>P</sup>* <sup>þ</sup> *<sup>H</sup>* (21)

ð Þ **x** *B*ð Þ **x** (22)

*a*ð Þ **x** will be obtained. So we have

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

with boundary conditions,

Then instead of *uj* from *U*, we can replace *u<sup>h</sup>*

where

equation

*<sup>L</sup>* <sup>X</sup>*<sup>N</sup> i*¼1

*<sup>N</sup>* <sup>X</sup>*<sup>N</sup> i*¼1

**63**

*ai*ð Þ **x** *u*1ð Þ **x***<sup>i</sup>* ,

*ai*ð Þ **x** *u*1ð Þ **x***<sup>i</sup>* ,

X*N i*¼1

> X*N i*¼1

*α*: is a small positive number that is selected experimentally.

Then in every node points, matrix A is computed.

By running the algorithm the new value is assigned to *λ*, this value is related to the condition number of matrix A and its amount will increase. Therefore, the size of the support domain is increased and then the matrix A with new nodal points in the support domain is reproduced. This loop is repeated until <sup>1</sup> *cond A*ð Þ <sup>≥</sup> *<sup>ε</sup>*.

The main idea of the moving least squares approximation is that for every point x can solve a locally weighted least squares problem [30], it is a local approximation, thus the additional condition to stop the loop is the size of the local support domain, the value of *λ* should be well enough to pave the local approximation, Line 20 is said to satisfy this condition.

#### **2.3 Modified MLS approximation method**

One of the common problems in Classic MLS method is the singularity of the moment matrix A in irregularity chosen nodal points. To avoid the nodal configurations which lead to a singular moment matrix, the usual solution is to enlarge the support domains of any nodal point. But it is not an appropriate solution, in [31] to tackle such problems is proposed a modified Moving least squares(MMLS)approximation method. This modifies allows, base functions in *m* ≥2 to be used with the same size of the support domain as linear base functions ð Þ *m* ¼ 1 *:* We should note that,impose additional terms based on the coefficients of the polynomial base functions is the main view of the modified technique. As we know, in the basis function **p x**ð Þ is

$$\mathbf{p}(\mathbf{x}) = \begin{bmatrix} \mathbf{1}, \boldsymbol{\varkappa}, \boldsymbol{\varkappa}^2, \dots, \boldsymbol{\varkappa}^m \end{bmatrix}^T \tag{15}$$

where **x**∈ , Also the correspond coefficients *aj*, that should be determined are [24]:

$$\mathbf{a(x)} = \begin{bmatrix} a\_1, a\_x, a\_{x^2}, \dots, a\_{x^m} \end{bmatrix}^T \tag{16}$$

For obtaining these coefficients, the functional (2) rewrite as follows:

$$\overline{J}(\mathbf{x}) = \sum\_{j=1}^{m} \left( \mathbf{P}^T(\mathbf{x}\_j) \mathbf{a}(\mathbf{x}) - u\_i(\mathbf{x}\_j) \right)^2 w\_i(\mathbf{x}) + \sum\_{\nu=1}^{m-2} \overline{w}\_{\nu}(\mathbf{x}) \overline{\mathbf{a}}\_{\nu}^2(\mathbf{x}), i = 1, 2, \dots, n \tag{17}$$

Where *w* is a vector of positive weights for the additional constraints, also **<sup>a</sup>** <sup>¼</sup> *ax*<sup>2</sup> , *ax*<sup>3</sup> , … , *ax* ½ � *<sup>m</sup> <sup>T</sup>* is the modified matrix.

The matrix form of (17) is as follows:

$$\tilde{J}(\mathbf{x}) = \left[P.\mathbf{a} - \mathbf{u}\_i\right]^T.\mathcal{W}.\left[P.\mathbf{a} - \mathbf{u}\_i\right] + \mathbf{a}^T H \mathbf{a}, i = 1, 2, \dots, n\tag{18}$$

where *H* is as,

$$H = \begin{bmatrix} O\_{2,2} & O\_{m-2,m-2} \\ O\_{2,2} & diag(\overline{w}) \end{bmatrix},\tag{19}$$

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

where, *Oi*,*<sup>j</sup>* is the null matrix. By minimizing the functional (18), the coefficients *a*ð Þ **x** will be obtained. So we have

$$
\overline{A}(\mathbf{x})\mathbf{a}(\mathbf{x}) = B(\mathbf{x})\mathbf{u}\_i,\tag{20}
$$

where

In Algorithm 1.

to satisfy this condition.

function **p x**ð Þ is

*<sup>J</sup>*ð Þ¼ **<sup>x</sup>** <sup>X</sup>*<sup>m</sup>*

*j*¼1

where *H* is as,

**62**

**P***<sup>T</sup>* **x***<sup>j</sup>*

� �**a x**ð Þ� *ui* **<sup>x</sup>***<sup>j</sup>* � � � � <sup>2</sup>

also **<sup>a</sup>** <sup>¼</sup> *ax*<sup>2</sup> , *ax*<sup>3</sup> , … , *ax* ½ � *<sup>m</sup> <sup>T</sup>* is the modified matrix. The matrix form of (17) is as follows:

*H* ¼

are [24]:

**2.3 Modified MLS approximation method**

*X*: is a set containing N scattered points which are called centers or data site and I

By running the algorithm the new value is assigned to *λ*, this value is related to the condition number of matrix A and its amount will increase. Therefore, the size of the support domain is increased and then the matrix A with new nodal points in

The main idea of the moving least squares approximation is that for every point x can solve a locally weighted least squares problem [30], it is a local approximation, thus the additional condition to stop the loop is the size of the local support domain, the value of *λ* should be well enough to pave the local approximation, Line 20 is said

One of the common problems in Classic MLS method is the singularity of the moment matrix A in irregularity chosen nodal points. To avoid the nodal configurations which lead to a singular moment matrix, the usual solution is to enlarge the support domains of any nodal point. But it is not an appropriate solution, in [31] to tackle such problems is proposed a modified Moving least squares(MMLS)approximation method. This modifies allows, base functions in *m* ≥2 to be used with the same size of the support domain as linear base functions ð Þ *m* ¼ 1 *:* We should note that,impose additional terms based on the coefficients of the polynomial base functions is the main view of the modified technique. As we know, in the basis

**p x**ð Þ¼ 1, *<sup>x</sup>*, *<sup>x</sup>*<sup>2</sup>

For obtaining these coefficients, the functional (2) rewrite as follows:

Where *w* is a vector of positive weights for the additional constraints,

*wi*ð Þþ **x**

*m* X�2 *ν*¼1

*<sup>J</sup>*ð Þ¼ **<sup>x</sup>** *<sup>P</sup>:***<sup>a</sup>** � **<sup>u</sup>***<sup>i</sup>* ½ �*<sup>T</sup>:W: <sup>P</sup>:***<sup>a</sup>** � **<sup>u</sup>***<sup>i</sup>* ½ �þ **<sup>a</sup>***TH***a**, *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* (18)

*O*2,2 *Om*�2,*m*�<sup>2</sup>

" #

*O*2,2 *diag*ð Þ *w*

where **x**∈ , Also the correspond coefficients *aj*, that should be determined

*cond A*ð Þ <sup>≥</sup> *<sup>ε</sup>*.

, … , *x<sup>m</sup>* � �*<sup>T</sup>* (15)

*<sup>ν</sup>*ð Þ **x** , *i* ¼ 1, 2, … , *n* (17)

, (19)

**a x**ð Þ¼ *<sup>a</sup>*1, *ax*, *ax*<sup>2</sup> , … , *ax* ½ � *<sup>m</sup> <sup>T</sup>* (16)

*<sup>w</sup>ν*ð Þ *<sup>x</sup>* **<sup>a</sup>**<sup>2</sup>

(x) is the Index of points which MLS shape function is evaluated. *α*: is a small positive number that is selected experimentally.

the support domain is reproduced. This loop is repeated until <sup>1</sup>

Then in every node points, matrix A is computed.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\overline{A} = P^T.W.P + H \tag{21}$$

And the matrics *B*ð Þ **x** is determined as the same before. So we have

$$\rho\_m(\mathbf{x}) = \mathbf{a}(\mathbf{x}) = p^T(\mathbf{x}) \overline{A}^{-1}(\mathbf{x}) B(\mathbf{x}) \tag{22}$$

where *φm*ð Þ **x** is the shape function of the MMLS approximation method.

#### **3. Stiff systems of ordinary differential equations**

In this section, we use MLS approximation method for numerical solution of the Stiff system of ordinary differential equations so consider the following differential equation

$$A(U) - F(\mathbf{x}) = \mathbf{0}, U(\mathbf{0}) = U\_0, \mathbf{x} \in \Omega \tag{23}$$

with boundary conditions,

$$B\left(U, \frac{\partial U}{\partial \mathbf{x}}\right) = \mathbf{0}, \mathbf{x} \in \partial \Omega.$$

where A is a general differential operator, *U*<sup>0</sup> is an initial approximation of (23), *F*ð Þ **x** is a vector of known analytical functions on the domain Ω and ∂Ω is the boundary of Ω. The operator can be divided by *A* ¼ *L* þ *N*, where *L* is the linear part, and *N* is the nonlinear part of its. So (23) can be, rewritten as follows;

$$L(U) + N(U) - F(\mathbf{x}) = \mathbf{0} \tag{24}$$

We assume that **a** ¼ f g *a*1, *a*2, … , *am* are the MLS shape functions so in order to solve system (24), *N* nodal points *xi* are selected on the Ω, which *xi* f g j*i* ¼ 1, 2, … , *N* is q-unisolvent. The distribution of nodes could be selected regularly or randomly. Then instead of *uj* from *U*, we can replace *u<sup>h</sup> <sup>j</sup>* from (13). So we have

$$u\_j^h(\mathbf{x}) = \sum\_{i=1}^N a\_i(\mathbf{x}) u\_j(\mathbf{x}\_i) \tag{25}$$

where *j* ¼ 1, 2, … , *n* is the number of unknown functions. we estimate the unknown functions *ui* by (25), so the system (24) becomes the following form

$$\begin{aligned} &L\left(\sum\_{i=1}^{N}a\_i(\mathbf{x})u\_1(\mathbf{x}\_i), \sum\_{i=1}^{N}a\_i(\mathbf{x})u\_2(\mathbf{x}\_i), \dots, \sum\_{i=1}^{N}a\_i(\mathbf{x})u\_n(\mathbf{x}\_i)\right) + \\ &N\left(\sum\_{i=1}^{N}a\_i(\mathbf{x})u\_1(\mathbf{x}\_i), \sum\_{i=1}^{N}a\_i(\mathbf{x})u\_2(\mathbf{x}\_i), \dots, \sum\_{i=1}^{N}a\_i(\mathbf{x})u\_n(\mathbf{x}\_i)\right) = \left(f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dots, f\_n(\mathbf{x})\right) + \mathbf{r}(\mathbf{x}). \end{aligned} \tag{26}$$

where **r x**ð Þ is residual error function which vanishes to zero in collocation points thus by installing the collocation points **y***r*;*r* ¼ 1, 2, … , *N*, so

the modified moving least squares approximation method it is sufficient to use

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

The convergence analysis of the method in matrix norm has been investigated for the systems of one and two-dimensional Fredholm integral equations by authors of [22]. It should be noted that The convergence analysis of the method for implementation classic moving least squares approximation method on a system of integral equations has been discussed and it should be investigated for modified Mls

So in continuation of this section, the error estimations for the system of differential equations is developed. In [26], has obtained error estimates for moving least square approximations in the one-dimensional case. Also in [33], is developed for functional in n-dimensional and was used the error estimates to prove an error estimate in Galerkin coercive problems. In this work, have improved error estimate

Given *δ*>0 let *W<sup>δ</sup>* ≥0 be a function such that *supp w*ð Þ*<sup>δ</sup>* ⊂*Bδ*ð Þ¼ 0 f g *z*k*z*j≤ *δ* and

*<sup>j</sup>*¼<sup>1</sup> is called a partition of unity subordinated to the open

� �, *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*. A class

*X<sup>δ</sup>* ¼ f g *x*1, *x*2, … , *xn* , *n* ¼ *n*ð Þ*δ* , a set of points in Ω ⊂ an open interval and let *U* ¼ ð Þ *u*1, *u*2, … , *uN* be the unknown function such that *ui*1, *ui*2, … , *uin* be the values

There is no unique way to build a partition of unity as defined above [34]. As we know, the MLS approximation is well defined if we have a unique solution at every point *x*∈ Ω. for minimization problem. And non-singularity of matrix *A x*ð Þ, ensuring it is. In [33] the error estimate was obtained with the following assumption

> *w x* � *xj* � �*u xj*

> > *w x* � *xj* � �*u xj*

<sup>∥</sup>*U*∥∞ <sup>¼</sup> *max ui* j j*x*, *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>* � �

� �*v xj* � �

� �<sup>2</sup>

method. we can use the results for this type of equations.

for the systems of stiff ordinary differential equations.

of the function *ui* in those points, i.e., *ui*,*<sup>j</sup>* ¼ *ui xj*

covering *IN* if it possesses the following properties:

about the system of nodes and weight functions f g Θ*N*,*WN* :

h i *<sup>u</sup>*, *<sup>v</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup>*

∥*u*∥<sup>2</sup>

Also for vector of unknown functions, we define

*j*¼1

*<sup>x</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> j*¼1

� �*<sup>N</sup>*

*<sup>s</sup>* , *s* >0 *or s* ¼ ∞,

*<sup>ω</sup><sup>j</sup>* <sup>¼</sup> <sup>1</sup> *for  every x*<sup>∈</sup> <sup>Ω</sup>

*φ<sup>m</sup>* instead of *φ:*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

**3.1 Error analysis**

of functions *W* ¼ *ω<sup>j</sup>*

� �⊆Λ*j*,

•*ωj*ð Þ *x* > 0, *x*∈Λ*j*,

• *Wj* ∈*C*<sup>0</sup>

•sup *ω<sup>j</sup>*

• X *N*

*i*¼1

We define

then

**65**

$$\begin{split} &L\left(\sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{1}(\mathbf{x}\_{i}), \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{2}(\mathbf{x}\_{i}), \dots, \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{n}(\mathbf{x}\_{i})\right) + \\ &N\left(\sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{1}(\mathbf{x}\_{i}), \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{2}(\mathbf{x}\_{i}), \dots, \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{n}(\mathbf{x}\_{i})\right) = \\ &\sum\_{i=1}^{N}L(a\_{i}(\mathbf{y}\_{r}))u\_{1}(\mathbf{x}\_{i}), \sum\_{i=1}^{N}L(a\_{i}(\mathbf{y}\_{r}))u\_{2}(\mathbf{x}\_{i}), \dots, \sum\_{i=1}^{N}L(a\_{i}(\mathbf{y}\_{r}))u\_{n}(\mathbf{x}\_{i})) + \\ &N\left(\sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{1}(\mathbf{x}\_{i}), \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{2}(\mathbf{x}\_{i}), \dots, \sum\_{i=1}^{N}a\_{i}(\mathbf{y}\_{r})u\_{n}(\mathbf{x}\_{i})\right) = \\ &(f\_{1}(\mathbf{y}\_{r}), f\_{2}(\mathbf{y}\_{r}), \dots, f\_{n}(\mathbf{y}\_{r})) \end{split} \tag{27}$$

therefore

$$\text{CU} = \begin{bmatrix} L(a\_1(\mathbf{y}\_1)) & L(a\_2(\mathbf{y}\_1)) & \dots & L(a\_N(\mathbf{y}\_1)) \\ L(a\_1(\mathbf{y}\_2)) & L(a\_2(\mathbf{y}\_2)) & \dots & L(a\_N(\mathbf{y}\_2)) \\ \vdots \\ L(a\_1(\mathbf{y}\_N)) & L(a\_2(\mathbf{y}\_N)) & \dots & L(a\_N(\mathbf{y}\_N)) \end{bmatrix} \begin{bmatrix} u\_1(\mathbf{x}\_1) & u\_2(\mathbf{x}\_1) & \dots & u\_n(\mathbf{x}\_1) \\ u\_1(\mathbf{x}\_2) & u\_2(\mathbf{x}\_2) & \dots & u\_n(\mathbf{x}\_2) \\ \vdots \\ u\_1(\mathbf{x}\_N) & u\_2(\mathbf{x}\_N) & \dots & u\_n(\mathbf{x}\_N) \end{bmatrix} \tag{28}$$

And the matrix form of (27) as follows

$$\mathbf{C}\_{\text{N}\times\text{N}}\mathbf{U}\_{\text{N}\times\text{n}} + \mathbf{N}\_{\text{N}\times\text{n}}(\mathbf{a}, \mathbf{U}) = \mathbf{F}\_{\text{N}\times\text{n}}(\mathbf{y}\_r) \tag{29}$$

where

$$\begin{aligned} \mathbf{C}\_{i} &= \left[ L\left( a\_{1}(\mathbf{y}\_{r}) \right), \dots, L\left( a\_{N}(\mathbf{y}\_{r}) \right) \right]\_{i=1}^{n} \\ U\_{i} &= \left[ \left( u\_{i}(\mathbf{x}\_{1}), u\_{i}(\mathbf{x}\_{2}), \dots, u\_{i}(\mathbf{x}\_{N}) \right)^{T} \right]\_{i=1}^{n} \\ F(\mathbf{y}\_{r}) &= \left( \left[ \left( f\_{1}(\mathbf{y}\_{r}) \right)\_{r=1}^{N} \right]^{T}, \left[ \left( f\_{2}(\mathbf{y}\_{r}) \right)\_{r=1}^{N} \right]^{T}, \dots \left[ \left( f\_{n}(\mathbf{y}\_{r}) \right)\_{r=1}^{N} \right] \right)^{T} . \end{aligned} \tag{30}$$

by imposing the initial conditions at *t* ¼ 0, we have

$$\left(\sum\_{i=1}^{N} a\_i(\mathbf{0}) u\_1(t\_i), \sum\_{i=1}^{N} a\_i(\mathbf{0}) u\_2(t\_i), \dots, \sum\_{i=1}^{N} a\_i(\mathbf{0}) u\_n(t\_i)\right) = U\_0 \tag{31}$$

and Solving the non-linear system (29) and (31), lead to finding quantities *uj*ð Þ *xi* . Then the values of *uj*ð Þ *x* at any point *x*∈ Ω, can be approximated by Eq. (25) as:

$$u\_j(\mathbf{x}) \simeq \sum\_{i=1}^N a\_i(\mathbf{x}) u\_j(\mathbf{x}\_i)$$

Remark

Note that, for simplicity, the modification derivation is made for bivariate data, but can be easily extended to higher dimensions. Also, for implementation, *Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

the modified moving least squares approximation method it is sufficient to use *φ<sup>m</sup>* instead of *φ:*

#### **3.1 Error analysis**

where **r x**ð Þ is residual error function which vanishes to zero in collocation points

X *N*

*ai* **y***<sup>r</sup>* � �*un*ð Þ **<sup>x</sup>***<sup>i</sup>*

*ai* **y***<sup>r</sup>* � �*un*ð Þ **<sup>x</sup>***<sup>i</sup>*

X *N*

*L ai* **y***<sup>r</sup>*

*i*¼1

*ai* **y***<sup>r</sup>* � �*un*ð Þ **<sup>x</sup>***<sup>i</sup>* þ

¼

(27)

(28)

(30)

� � � � *un*ð ÞÞþ **<sup>x</sup>***<sup>i</sup>*

¼

*u*1ð Þ *x*<sup>1</sup> *u*2ð Þ *x*<sup>1</sup> … *un*ð Þ *x*<sup>1</sup> *u*1ð Þ *x*<sup>2</sup> *u*2ð Þ *x*<sup>2</sup> … *un*ð Þ *x*<sup>2</sup>

*u*1ð Þ *xN u*2ð Þ *xN* … *un*ð Þ *xN*

� � (29)

*T :*

¼ *U*<sup>0</sup> (31)

*i*¼1

X *N*

*i*¼1

X *N*

*i*¼1

⋮

*i*¼1

, … *f <sup>n</sup>* **y***<sup>r</sup>* � � � � *<sup>N</sup>*

X *N*

*i*¼1

*ai*ð Þ **x** *uj*ð Þ **x***<sup>i</sup>*

*i*¼1

*ai*ð Þ 0 *un*ð Þ*ti*

*r*¼1

� � � �

� � � �

� � � �

*CN*�*NUN*�*<sup>n</sup>* þ *NN*�*<sup>n</sup>*ð Þ¼ **a**, *U FN*�*<sup>n</sup> yr*

� � � � , … , *L aN* **<sup>y</sup>***<sup>r</sup>* � � � � � � *<sup>n</sup>*

*<sup>T</sup>* h i*<sup>n</sup>*

� � h i

*ai*ð Þ 0 *u*2ð Þ*ti* , … ,

and Solving the non-linear system (29) and (31), lead to finding quantities *uj*ð Þ *xi* . Then the values of *uj*ð Þ *x* at any point *x*∈ Ω, can be approximated by Eq. (25) as:

!

*uj*ð Þ **<sup>x</sup>** <sup>≃</sup> <sup>X</sup> *N*

*i*¼1

Note that, for simplicity, the modification derivation is made for bivariate data, but can be easily extended to higher dimensions. Also, for implementation,

*r*¼1 h i*<sup>T</sup>*

*Ui* ¼ ð Þ *ui*ð Þ *x*<sup>1</sup> , *ui*ð Þ *x*<sup>2</sup> , … , *ui*ð Þ *xN*

, *f* <sup>2</sup> **y***<sup>r</sup>* � � � � *<sup>N</sup>*

thus by installing the collocation points **y***r*;*r* ¼ 1, 2, … , *N*, so

*ai* **y***<sup>r</sup>*

*ai* **y***<sup>r</sup>*

*L ai* **y***<sup>r</sup>*

!

� � � � … *L aN <sup>y</sup>*<sup>1</sup>

� � � � … *L aN <sup>y</sup>*<sup>2</sup>

� � � � … *L aN yN*

*ai* **y***<sup>r</sup>*

� �*u*2ð Þ **<sup>x</sup>***<sup>i</sup>* , … ,

� �*u*2ð Þ **<sup>x</sup>***<sup>i</sup>* , … ,

� �*u*2ð Þ **<sup>x</sup>***<sup>i</sup>* , … ,

� � � � *<sup>u</sup>*2ð Þ **<sup>x</sup>***<sup>i</sup>* , … ,

!

!

X *N*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*i*¼1

X *N*

*i*¼1

X *N*

*i*¼1

X *N*

*i*¼1

*L* X *N*

*N* X *N*

X *N*

*i*¼1

*N* X *N*

> *f* <sup>1</sup> **y***<sup>r</sup>* � �, *<sup>f</sup>* <sup>2</sup> *yr*

therefore

where

*F* **y***<sup>r</sup>*

Remark

**64**

� � <sup>¼</sup> *<sup>f</sup>* <sup>1</sup> **<sup>y</sup>***<sup>r</sup>*

X *N*

*i*¼1

*CU* ¼

*i*¼1

*L a*<sup>1</sup> *y*<sup>1</sup>

*L a*<sup>1</sup> *y*<sup>2</sup>

⋮ *L a*<sup>1</sup> *yN*

*i*¼1

*i*¼1

*L ai* **y***<sup>r</sup>*

*ai* **y***<sup>r</sup>*

*ai* **y***<sup>r</sup>*

*ai* **y***<sup>r</sup>*

� �*u*1ð Þ **<sup>x</sup>***<sup>i</sup>* ,

� �*u*1ð Þ **<sup>x</sup>***<sup>i</sup>* ,

� �*u*1ð Þ *xi* ,

� � � � *L a*<sup>2</sup> *<sup>y</sup>*<sup>1</sup>

� � � � *L a*<sup>2</sup> *<sup>y</sup>*<sup>2</sup>

� � � � *L a*<sup>2</sup> *yN*

And the matrix form of (27) as follows

*Ci* ¼ *L a*<sup>1</sup> **y***<sup>r</sup>*

*r*¼1 h i*<sup>T</sup>*

by imposing the initial conditions at *t* ¼ 0, we have

X *N*

*i*¼1

� � � � *<sup>N</sup>*

*ai*ð Þ 0 *u*1ð Þ*ti* ,

� �, … , *<sup>f</sup> <sup>n</sup>* **<sup>y</sup>***<sup>r</sup>* � � � �

� � � � *<sup>u</sup>*1ð Þ **<sup>x</sup>***<sup>i</sup>* ,

The convergence analysis of the method in matrix norm has been investigated for the systems of one and two-dimensional Fredholm integral equations by authors of [22]. It should be noted that The convergence analysis of the method for implementation classic moving least squares approximation method on a system of integral equations has been discussed and it should be investigated for modified Mls method. we can use the results for this type of equations.

So in continuation of this section, the error estimations for the system of differential equations is developed. In [26], has obtained error estimates for moving least square approximations in the one-dimensional case. Also in [33], is developed for functional in n-dimensional and was used the error estimates to prove an error estimate in Galerkin coercive problems. In this work, have improved error estimate for the systems of stiff ordinary differential equations.

Given *δ*>0 let *W<sup>δ</sup>* ≥0 be a function such that *supp w*ð Þ*<sup>δ</sup>* ⊂*Bδ*ð Þ¼ 0 f g *z*k*z*j≤ *δ* and *X<sup>δ</sup>* ¼ f g *x*1, *x*2, … , *xn* , *n* ¼ *n*ð Þ*δ* , a set of points in Ω ⊂ an open interval and let *U* ¼ ð Þ *u*1, *u*2, … , *uN* be the unknown function such that *ui*1, *ui*2, … , *uin* be the values of the function *ui* in those points, i.e., *ui*,*<sup>j</sup>* ¼ *ui xj* � �, *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*. A class of functions *W* ¼ *ω<sup>j</sup>* � �*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup> is called a partition of unity subordinated to the open covering *IN* if it possesses the following properties:

• *Wj* ∈*C*<sup>0</sup> *<sup>s</sup>* , *s* >0 *or s* ¼ ∞, •sup *ω<sup>j</sup>* � �⊆Λ*j*, •*ωj*ð Þ *x* > 0, *x*∈Λ*j*, • X *N i*¼1 *<sup>ω</sup><sup>j</sup>* <sup>¼</sup> <sup>1</sup> *for  every x*<sup>∈</sup> <sup>Ω</sup>

There is no unique way to build a partition of unity as defined above [34].

As we know, the MLS approximation is well defined if we have a unique solution at every point *x*∈ Ω. for minimization problem. And non-singularity of matrix *A x*ð Þ, ensuring it is. In [33] the error estimate was obtained with the following assumption about the system of nodes and weight functions f g Θ*N*,*WN* :

We define

$$\langle u, v \rangle = \sum\_{j=1}^{n} w\left(\boldsymbol{x} - \boldsymbol{x}\_{j}\right) \boldsymbol{u}\left(\boldsymbol{x}\_{j}\right) \boldsymbol{v}\left(\boldsymbol{x}\_{j}\right),$$

then

$$\|\mathfrak{u}\|\_{\mathfrak{x}}^2 = \sum\_{j=1}^n w\left(\mathfrak{x} - \mathfrak{x}\_j\right) \mathfrak{u}\left(\mathfrak{x}\_j\right)^2$$

Also for vector of unknown functions, we define

$$\|U\|\_{\infty} = \max\left\{ |\mu\_i|\_{\infty}, i = 1, 2, \dots, N \right\},$$

are the discrete norm on the polynomial space <sup>1</sup> *<sup>m</sup>* if the weight function *w* satisfy the following properties.

**3.2 System of ODE**

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

And from (25), we define

*N*

operators,

where ð Þ *ai*

And demand that

operations.

*Lh*ð Þ¼ *<sup>U</sup> F we have*

**Proof.** we have

<sup>∥</sup>*LUt* ð Þ� ð Þ *<sup>L</sup><sup>h</sup>*

so that

**67**

If in (24) the non-linear operator *N* be zero, we have

*L U*ð Þ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>n</sup>*

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

where *U* is the vector of unknown function and *L* is a matrix of derivative

*ς*¼1 *λς ∂ς*

*i*¼1

*ai*ð Þ*t U t*ð Þ*<sup>i</sup>*

*<sup>i</sup>*¼<sup>1</sup> are the MLs shape functions defined on the interval 0, 1 ½ � satisfying

*L U*ð Þ¼ ð Þ*:* <sup>X</sup>*<sup>n</sup>*

ðÞ¼ *<sup>t</sup>* <sup>X</sup> *N*

the homogeneous counterparts of the boundary conditions in (23). Also if the weight function *w* possesses *k* continuous derivatives then the shape functions *aj* is also in *<sup>C</sup><sup>k</sup>* [33]. By the collocation method, is obtained an approximate solution *<sup>U</sup><sup>h</sup>*ð Þ*<sup>t</sup>* .

ð Þ¼ *<sup>U</sup>*ð Þ*:* <sup>X</sup>*<sup>n</sup>*

*ς*¼0 *λς ∂ς*

where (*λ* ¼ 0 *or* 1). It is assumed that in the system of ODE derivative of order at most *n* ¼ 2. Each of the basis functions *ai* has compact support contained in 0, 1 ð Þ then the matrix *C* in (30) is a bounded matrix. If *δ* be fixed, independent of *N*, then the resulting system of linear equations can be solved in *O N*ð Þ arithmetic

*m* ≥ 1 *and* ∥*ui*∥∞ ¼ *uk*, *k*∈ f g 1, 2, … , *n where* Ω *be a closed, bounded set in R. Assume the quadrature scheme is convergent for all continuous functions on* Ω*: Further, assume that the stiff system of ODE (23) with the fixd initial condition is uniquely solvable for given fi* <sup>∈</sup>*C*ð Þ <sup>Ω</sup> *: Moreover take a suitable approximation U<sup>h</sup> of U Then for all sufficiently large n, the approximate matrix L for linearly case exist and are uniformly bounded,* ∣*L*∣ ≤ *M with a suitable constant M* < ∞*: For the equations L U*ð Þ¼ *F and*

*Et* <sup>¼</sup> <sup>∥</sup>*LUt* ð Þ� ð Þ *Lh*ð Þ *U t*ð Þ ∥∞

<sup>∥</sup>*Et*∥∞ <sup>≤</sup> <sup>C</sup>*qK*ð Þ *<sup>λ</sup>*, *<sup>ς</sup> <sup>R</sup><sup>m</sup>*þ1�*<sup>μ</sup>*∥*u*ð Þ *<sup>m</sup>*þ<sup>1</sup>

X*n ς*¼0 *λς ∂ς*

ð Þ *U t*ð Þ ∥∞ ¼ ∥

*Uh*

*Lh*

**Lemma 3.2.** *Let U* ¼ ð Þ *u*1, *u*2, … *un and F* ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … *f <sup>n</sup>*

� � (33)

ð Þ*: <sup>ς</sup> <sup>U</sup>*ð Þ*: :* (34)

ð Þ*: <sup>ς</sup> <sup>U</sup><sup>h</sup>*ð Þ*:* (35)

� � *so that ui* ∈*C<sup>m</sup>*þ<sup>1</sup> Ω

*<sup>k</sup>* ∥*<sup>L</sup>*<sup>∞</sup> *:*

*ς*¼0 *λς ∂ς*

*<sup>t</sup><sup>ς</sup> <sup>U</sup><sup>h</sup>*ð Þ*<sup>t</sup>* ∥∞

*<sup>t</sup><sup>ς</sup> U t*ðÞ�X*<sup>n</sup>*

� �

**a**. For each *x*∈ Ω, *w x* � *xj* <sup>&</sup>gt;0 at least for ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> indices *<sup>j</sup>*.

**<sup>b</sup>**. For any *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup>, the moment matrix *A x*ð Þ¼ *w x*ð Þ*PT* is nonsingular.

**Definition 3.1.** *Given <sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>*, the set ST*ð Þ¼ *<sup>x</sup> <sup>j</sup>* : *<sup>ω</sup><sup>j</sup>* 6¼ <sup>0</sup> *will be called the star of x.* **Theorem 3.1.** *[34, 35] A necessary condition for the satisfaction of Property b is that for any x*∈ Ω

$$m = \operatorname{card}(ST(\mathfrak{x})) \ge \operatorname{card}(\mathbb{P}\_m) = m + 1$$

For a sample point **c** ∈ Ω, if *ST*ð Þ¼ **c** *j* <sup>1</sup>, … *j k* , the mesh-size of the star *ST*ð Þ**<sup>c</sup>** defined by the number is *h ST* ð Þ¼ ð Þ**<sup>c</sup>** max *hj*1, … *hjk :*

**Assumptions.** Consider the following global assumptions about parameters. There exist

ð Þ *a*<sup>1</sup> An over bound of the overlap of clouds:

$$E = \sup\_{c \in \overline{\Omega}} \{ card(ST(c))\}.$$

ð Þ *a*<sup>2</sup> Upper bounds of the condition number:

$$\mathsf{CB}\_q = \sup\_{c \in \overline{\Omega}} \{ \mathsf{CN}\_q(\mathsf{ST}(c)), q = 1, 2 \}.$$

where the numbers *CNq*ð Þ *ST*ð Þ**c** are computable measures of the quality of the star *ST c*ð Þ which defined in Theorem 7 of [19].

ð Þ *a*<sup>3</sup> An upper bound of the mesh-size of stars:

$$R = \sup\_{c \in \overline{\Omega}} (hST(c)).$$

ð Þ *a*<sup>4</sup> An uniform bound of the derivatives of *wj :* That is the constant *Gq* >0, *q* ¼ 1, 2, such that

$$\left\| D^{\mu} W\_{\hat{f}} \right\|\_{L\_{\infty}} \leq \frac{G\_q}{R^{|\mu|}} \quad \mathbf{1} < \mu < q,$$

ð Þ *a*<sup>5</sup> There exist the number *γ* >0 such that any two points **x**, **y**∈ Ω can be joined by a rectifiable curve <sup>Γ</sup> in <sup>Ω</sup> with length <sup>∣</sup>Γ∣ ≤*<sup>γ</sup>* **<sup>x</sup>**‐**<sup>y</sup>** *:*

Assuming all these conditions, Zuppa [34] proved.

**Lemma 3.1.** *<sup>U</sup>* <sup>¼</sup> ð Þ *<sup>u</sup>*1, *<sup>u</sup>*2, … *un such that ui* <sup>∈</sup>*C<sup>m</sup>*þ<sup>1</sup> <sup>Ω</sup> *and U*k k<sup>∞</sup> <sup>¼</sup> *uk*, 1<*k*<*n, There exist constants Cq*, *q* ¼ 1 *or* 2,

$$\mathcal{C}\_1 = \mathcal{C}\_1(\boldsymbol{\gamma}, \boldsymbol{d}, \boldsymbol{E}, \mathcal{G}\_1, \mathcal{C}\mathcal{B}\_1),$$

$$\mathcal{C}\_2 = \mathcal{C}\_1(\boldsymbol{\gamma}, \boldsymbol{d}, \boldsymbol{E}, \mathcal{G}\_2, \mathcal{C}\mathcal{B}\_1, \mathcal{C}\mathcal{B}\_2),$$

then

$$\left\| \left| D^{\mu} U - D^{\mu} U^{h} \right| \right\|\_{\infty} < \mathcal{C}\_{q} R^{q+1-|\mu|} \left\| u\_{k}^{(m+1)} \right\|\_{L^{\infty}(\Omega)} \quad \mathbf{0} < \mu < q \tag{32}$$

**Proof:** see [36].

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

#### **3.2 System of ODE**

are the discrete norm on the polynomial space <sup>1</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

<sup>&</sup>gt;0 at least for ð Þ *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> indices *<sup>j</sup>*.

**Definition 3.1.** *Given <sup>x</sup>* <sup>∈</sup> <sup>Ω</sup>*, the set ST*ð Þ¼ *<sup>x</sup> <sup>j</sup>* : *<sup>ω</sup><sup>j</sup>* 6¼ <sup>0</sup> *will be called the star of x.* **Theorem 3.1.** *[34, 35] A necessary condition for the satisfaction of Property b is that*

*n* ¼ *card ST* ð Þ ð Þ *x* ≥ *card*ð Þ¼ *<sup>m</sup> m* þ 1

**Assumptions.** Consider the following global assumptions about parameters.

*E* ¼ sup*<sup>c</sup>*<sup>∈</sup> <sup>Ω</sup>f g *card ST c* ð Þ ð Þ *:*

*CBq* <sup>¼</sup> sup*<sup>c</sup>*<sup>∈</sup> <sup>Ω</sup> *CNq*ð Þ *ST c*ð Þ , *<sup>q</sup>* <sup>¼</sup> 1, 2 *:*

where the numbers *CNq*ð Þ *ST*ð Þ**c** are computable measures of the quality of the

*R* ¼ sup*<sup>c</sup>*<sup>∈</sup> <sup>Ω</sup>ð Þ *hST c*ð Þ *:*

*Gq*

ð Þ *a*<sup>5</sup> There exist the number *γ* >0 such that any two points **x**, **y**∈ Ω can be joined

*C*<sup>1</sup> ¼ *C*1ð Þ *γ*, *d*, *E*, *G*1,*CB*<sup>1</sup> ,

*C*<sup>2</sup> ¼ *C*1ð Þ *γ*, *d*, *E*, *G*2,*CB*1,*CB*<sup>2</sup> ,

<sup>∞</sup> <sup>&</sup>lt;*CqR<sup>q</sup>*þ1�∣*μ*<sup>∣</sup>

*<sup>R</sup>*∣*μ*<sup>∣</sup> <sup>1</sup><*μ*<*q*,

 *:*

∥*u*ð Þ *<sup>m</sup>*þ<sup>1</sup>

<sup>1</sup>, … *j k*

*:*

**<sup>b</sup>**. For any *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup>, the moment matrix *A x*ð Þ¼ *w x*ð Þ*PT* is nonsingular.

satisfy the following properties.

**a**. For each *x*∈ Ω, *w x* � *xj*

For a sample point **c** ∈ Ω, if *ST*ð Þ¼ **c** *j*

defined by the number is *h ST* ð Þ¼ ð Þ**c** max *hj*1, … *hjk*

ð Þ *a*<sup>1</sup> An over bound of the overlap of clouds:

ð Þ *a*<sup>2</sup> Upper bounds of the condition number:

star *ST c*ð Þ which defined in Theorem 7 of [19]. ð Þ *a*<sup>3</sup> An upper bound of the mesh-size of stars:

*Gq* >0, *q* ¼ 1, 2, such that

*There exist constants Cq*, *q* ¼ 1 *or* 2,

then

**66**

**Proof:** see [36].

ð Þ *a*<sup>4</sup> An uniform bound of the derivatives of *wj*

*Dμ Wj*

 *<sup>L</sup>*<sup>∞</sup> ≤

Assuming all these conditions, Zuppa [34] proved. **Lemma 3.1.** *<sup>U</sup>* <sup>¼</sup> ð Þ *<sup>u</sup>*1, *<sup>u</sup>*2, … *un such that ui* <sup>∈</sup>*C<sup>m</sup>*þ<sup>1</sup> <sup>Ω</sup>

by a rectifiable curve <sup>Γ</sup> in <sup>Ω</sup> with length <sup>∣</sup>Γ∣ ≤*<sup>γ</sup>* **<sup>x</sup>**‐**<sup>y</sup>**

*<sup>D</sup><sup>μ</sup><sup>U</sup>* � *<sup>D</sup><sup>μ</sup>U<sup>h</sup>* 

*for any x*∈ Ω

There exist

*<sup>m</sup>* if the weight function *w*

, the mesh-size of the star *ST*ð Þ**<sup>c</sup>**

*:* That is the constant

*and U*k k<sup>∞</sup> <sup>¼</sup> *uk*, 1<*k*<*n,*

*<sup>k</sup>* <sup>∥</sup>*L*∞ð Þ <sup>Ω</sup> <sup>0</sup><*μ*<*<sup>q</sup>* (32)

If in (24) the non-linear operator *N* be zero, we have

$$L(U) = \begin{pmatrix} f\_1, f\_2, \dots, f\_n \end{pmatrix} \tag{33}$$

where *U* is the vector of unknown function and *L* is a matrix of derivative operators,

$$L(U(.)) = \sum\_{\xi=1}^{n} \lambda\_{\xi} \frac{\partial^{\xi}}{(.)^{\xi}} U(.). \tag{34}$$

And from (25), we define

$$U^h(t) = \sum\_{i=1}^N a\_i(t)U(t\_i)$$

where ð Þ *ai N <sup>i</sup>*¼<sup>1</sup> are the MLs shape functions defined on the interval 0, 1 ½ � satisfying the homogeneous counterparts of the boundary conditions in (23). Also if the weight function *w* possesses *k* continuous derivatives then the shape functions *aj* is also in *<sup>C</sup><sup>k</sup>* [33]. By the collocation method, is obtained an approximate solution *<sup>U</sup><sup>h</sup>*ð Þ*<sup>t</sup>* . And demand that

$$L^h(U(.)) = \sum\_{\varsigma=0}^n \lambda\_\varsigma \frac{\partial^\varsigma}{(.)^\varsigma} U^h(.)\tag{35}$$

where (*λ* ¼ 0 *or* 1). It is assumed that in the system of ODE derivative of order at most *n* ¼ 2. Each of the basis functions *ai* has compact support contained in 0, 1 ð Þ then the matrix *C* in (30) is a bounded matrix. If *δ* be fixed, independent of *N*, then the resulting system of linear equations can be solved in *O N*ð Þ arithmetic operations.

**Lemma 3.2.** *Let U* ¼ ð Þ *u*1, *u*2, … *un and F* ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … *f <sup>n</sup>* � � *so that ui* ∈*C<sup>m</sup>*þ<sup>1</sup> Ω � � *m* ≥ 1 *and* ∥*ui*∥∞ ¼ *uk*, *k*∈ f g 1, 2, … , *n where* Ω *be a closed, bounded set in R. Assume the quadrature scheme is convergent for all continuous functions on* Ω*: Further, assume that the stiff system of ODE (23) with the fixd initial condition is uniquely solvable for given fi* <sup>∈</sup>*C*ð Þ <sup>Ω</sup> *: Moreover take a suitable approximation U<sup>h</sup> of U Then for all sufficiently large n, the approximate matrix L for linearly case exist and are uniformly bounded,* ∣*L*∣ ≤ *M with a suitable constant M* < ∞*: For the equations L U*ð Þ¼ *F and Lh*ð Þ¼ *<sup>U</sup> F we have*

$$E\_t = \|L(U(t)) - L^h(U(t))\|\_\infty$$

so that

$$\|\|E\_t\|\|\_{\infty} \le \mathcal{C}\_q K(\lambda, \mathfrak{z}) R^{m+1-\mu} \|\|u\_k^{(m+1)}\|\|\_{L\_{\infty}}.$$

**Proof.** we have

$$\|\|L(U(t)) - L^h(U(t))\|\|\_{\infty} = \|\sum\_{\xi=0}^n \lambda\_{\xi} \frac{\partial^{\xi}}{t^{\xi}} U(t) - \sum\_{\varsigma=0}^n \lambda\_{\varsigma} \frac{\partial^{\varsigma}}{t^{\varsigma}} U^h(t) \|\_{\infty}$$

so due to the lemma (36),

$$\begin{aligned} \|L(U(t)) - L^h(U(t))\|\_{\infty} &\le \sum\_{\varsigma=0}^n |\lambda\_{\varsigma}| \|\frac{\partial^{\varsigma}}{t^{\varsigma}} U(t) - \frac{\partial^{\varsigma}}{t^{\varsigma}} U^h(t)\|\_{\infty} \\ &\le \max\_i \sum\_{\varsigma=0}^n |\lambda\_{\varsigma}| \|\frac{\partial^{\varsigma}}{t^{\varsigma}} u\_i(t) - \frac{\partial^{\varsigma}}{t^{\varsigma}} u\_i^h(t)\|\_{\infty} \\ &\le \sum\_{\varsigma=0}^n \mathcal{C}\_q |\lambda\_{\varsigma}| \|u\_k^{(m+1)}\|\_{L\_{\infty}} R^{m+1-\varsigma} \end{aligned}$$

<sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>θ</sup>*, *<sup>s</sup>* <sup>∈</sup> <sup>2</sup> : �1≤*s*<sup>≤</sup> 1, *<sup>g</sup>*1ð Þ*<sup>s</sup>* <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup>*g*2ð Þ*<sup>s</sup>* � � (37)

<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>g</sup>*2ð Þ� *<sup>θ</sup> <sup>g</sup>*1ð Þ*<sup>θ</sup>*

Kð Þ *x*, *y*, *t*, *s* **U**ð Þ *t*, *s dtds*, *such that x*ð Þ , *y* ∈½ �� � �1, 1 ½ � 1, 1

the interval *<sup>g</sup>*1ð Þ*<sup>θ</sup>* , *<sup>g</sup>*2ð Þ*<sup>θ</sup>* � � is converted to the fixed interval ½ � �1, 1 , so we have

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

<sup>2</sup> , (38)

Kð Þ *x*, *y*, *θ*, *s* (40)

*<sup>l</sup>*¼<sup>1</sup>Ω*<sup>l</sup>* and <sup>Ω</sup>*<sup>l</sup>* <sup>⋂</sup> <sup>Ω</sup>*<sup>k</sup>* 6¼ <sup>∅</sup>, 1≤*k*, *<sup>l</sup>* <sup>≤</sup>*<sup>L</sup>*

*<sup>n</sup>*ð Þ **<sup>x</sup>** � �*<sup>T</sup>* (42)

<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>s</sup>* , *<sup>u</sup><sup>h</sup>*

*g s*ð Þ*ds* (41)

� � (43)

<sup>2</sup>ð Þ *<sup>t</sup>*, *<sup>s</sup>* , … *<sup>u</sup><sup>h</sup> <sup>n</sup>*ð Þ *<sup>t</sup>*, *<sup>s</sup>* � �*<sup>T</sup>*

*dtds:*

(44)

(39)

by the following linear transformation

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

ð 1

ð 1

�1

�1

So, for the numerical integration <sup>Ω</sup> <sup>¼</sup> <sup>⋃</sup>*<sup>L</sup>*

*<sup>i</sup>* from (12). So we have

Also, obviously from (12)

<sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *uh <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>* <sup>¼</sup> *f x*ð Þþ , *<sup>y</sup>*

becomes as follows

*uh* <sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *<sup>u</sup><sup>h</sup>*

**69**

ð

Ω

*<sup>U</sup><sup>h</sup>*ð Þ¼ **<sup>x</sup>** *uh*

*uh*

*<sup>j</sup>* ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* <sup>X</sup>

**U**ð Þ¼ *x*, *y* **F**ð Þþ *x*, *y*

where

earlier.

½ �� *a*, *b* ½ � *c*, *d* .

replace *u<sup>h</sup>*

*<sup>θ</sup>*ð Þ¼ *<sup>t</sup>*, *<sup>s</sup> <sup>g</sup>*2ð Þ� *<sup>θ</sup> <sup>g</sup>*1ð Þ*<sup>θ</sup>*

<sup>K</sup>ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup> <sup>g</sup>*2ð Þ� *<sup>θ</sup> <sup>g</sup>*1ð Þ*<sup>θ</sup>*

2

*L*

ð

Ω*l*

<sup>2</sup>ð Þ **<sup>x</sup>** , … , *uh*

*ϕi*ð Þ *x*, *y uj xi*, *yi*

*kij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>* � � *<sup>u</sup><sup>h</sup>*

*l*¼1

Also, the second kind is straight similarly by commuting the variables and the third kind can be separated to finite numbers of sub-domains of the first or second kinds, so the method can be applied in each sub-domain as described

*g s*ð Þ*ds* <sup>¼</sup> <sup>X</sup>

Here, the MLS method is applied for the general case where the domain is

<sup>1</sup> ð Þ **<sup>x</sup>** , *<sup>u</sup><sup>h</sup>*

*N*

*i*¼1

ð *d* ð *b*

*a*

*c*

in this section, is assumed that the domain has rectangular shape, so system (36)

To apply the method, as described in section 2.1, instead of *ui* from *U*, we can

where should be *m* ≥ *ς* so,

$$\sum\_{\varsigma=0}^{n} |\lambda\_{\varsigma}| R^{m+1-\varsigma} \le K(\lambda, \varsigma) R^{m+1-\mu}$$

where *<sup>μ</sup>* is the highest order derivative And *<sup>K</sup>*ð Þ¼ *<sup>λ</sup>*, *<sup>ς</sup>* <sup>P</sup>*<sup>n</sup> ς*¼0 ∣*λς*∣, so demanded that

$$\|\|E\_t\|\|\_{\infty} \le \mathcal{C}\_q K(\lambda, \mathfrak{z}) R^{m+1-\mu} \|\|u\_k^{(m+1)}\|\|\_{L\_{\infty}}.$$

It should be noted that in the nonlinear system the upper bound of error depends on the nonlinear operator.

#### **4. Two-dimensional linear systems of integral equations**

#### **4.1 Fredholm type**

In this section, we will provide an approximation solution of the 2-D linear system of Fredholm integral equations by the MLS method. The matrix form of this system could be considered as

$$\mathbf{U}(\mathbf{x},\boldsymbol{y}) = \mathbf{F}(\mathbf{x},\boldsymbol{y}) + \int\_{\Omega} \mathbf{K}(\mathbf{x},\boldsymbol{y},\theta,\mathbf{s}) \mathbf{U}(\theta,\mathbf{s}) d\theta d\mathbf{s}, \quad (\mathbf{x},\boldsymbol{y}) \in \Omega,\tag{36}$$

where <sup>Ω</sup> <sup>¼</sup> ½ �� *<sup>a</sup>*, *<sup>b</sup>* ½ � *<sup>c</sup>*, *<sup>d</sup>* as <sup>Ω</sup> <sup>⊂</sup> <sup>2</sup> , Also *K x*ð Þ¼ , *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup> <sup>κ</sup>ij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup>* � �, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *n* is the matrix of kernels, **U**ð Þ¼ *x*, *y* ð Þ *u*1ð Þ *x*, *y* , *u*2ð Þ *x*, *y* , … *un*ð Þ *x*, *y <sup>T</sup>* is the vector of unknown function and **<sup>F</sup>**ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>f</sup>* <sup>1</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *<sup>f</sup> <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>* is the vector of known functions.

In addition, is took that two cases for the domain, the rectangular shape, and nonrectangular one and three cases relative to the geometry of the nonrectangular domain are considered where can be transformed into the rectangular shape [35].

The first one is <sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>θ</sup>*, *<sup>s</sup>* <sup>∈</sup> <sup>2</sup> : *<sup>a</sup>*<sup>≤</sup> *<sup>s</sup>*≤*b*, *<sup>g</sup>*1ð Þ*<sup>s</sup>* <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup> *<sup>g</sup>*2ð Þ*<sup>s</sup>* � � where *<sup>g</sup>*1ð Þ*<sup>s</sup>* and *g*2ð Þ*s* are continues functions of *s*, the second one can be consider as Ω ¼ ð Þ *<sup>θ</sup>*, *<sup>s</sup>* <sup>∈</sup> <sup>2</sup> : *<sup>c</sup>*<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>*d*, *<sup>g</sup>*1ð Þ*<sup>θ</sup>* <sup>≤</sup> *<sup>s</sup>*≤*g*2ð Þ*<sup>θ</sup>* � � where *<sup>g</sup>*1ð Þ*<sup>θ</sup>* and *<sup>g</sup>*2ð Þ*<sup>θ</sup>* are continues functions of *θ*, Also the last one is a domain which is neither of the first nor the second kinds but could be separated to finite numbers of first or second subdomains, it is labeled as a domain of third kind. Without loss of generality, the first kind domain can be assumed as

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

$$\mathfrak{Q} = \left\{ (\theta, s) \in \mathbb{R}^2 \, : \, -\mathbf{1} \le s \le \mathbf{1}, \mathbf{g}\_1(s) \le \theta \le \mathbf{g}\_2(s) \right\} \tag{37}$$

by the following linear transformation

$$\theta(t,s) = \frac{\mathbf{g}\_2(\theta) - \mathbf{g}\_1(\theta)}{2}t + \frac{\mathbf{g}\_2(\theta) - \mathbf{g}\_1(\theta)}{2},\tag{38}$$

the interval *<sup>g</sup>*1ð Þ*<sup>θ</sup>* , *<sup>g</sup>*2ð Þ*<sup>θ</sup>* � � is converted to the fixed interval ½ � �1, 1 , so we have

$$\mathbf{U}(\mathbf{x},\boldsymbol{\uprho}) = \mathbf{F}(\mathbf{x},\boldsymbol{\uprho}) + \int\_{-1}^{1} \int\_{-1}^{1} \mathbf{K}(\mathbf{x},\boldsymbol{\uprho},t,s)\mathbf{U}(t,s)dtds,\ such\ that(\mathbf{x},\boldsymbol{\uprho}) \in [-1,1] \times [-1,1] \tag{39}$$

where

so due to the lemma (36),

where should be *m* ≥ *ς* so,

on the nonlinear operator.

system could be considered as

vector of known functions.

**U**ð Þ¼ *x*, *y* **F**ð Þþ *x*, *y*

where <sup>Ω</sup> <sup>¼</sup> ½ �� *<sup>a</sup>*, *<sup>b</sup>* ½ � *<sup>c</sup>*, *<sup>d</sup>* as <sup>Ω</sup> <sup>⊂</sup> <sup>2</sup>

first kind domain can be assumed as

**68**

**4.1 Fredholm type**

<sup>∥</sup>*LUt* ð Þ� ð Þ *Lh*

ð Þ *U t*ð Þ ∥∞ <sup>≤</sup> <sup>X</sup>*<sup>n</sup>*

X*n ς*¼0

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

where *<sup>μ</sup>* is the highest order derivative And *<sup>K</sup>*ð Þ¼ *<sup>λ</sup>*, *<sup>ς</sup>* <sup>P</sup>*<sup>n</sup>*

**4. Two-dimensional linear systems of integral equations**

ð

Ω

1, 2, … , *n* is the matrix of kernels, **U**ð Þ¼ *x*, *y* ð Þ *u*1ð Þ *x*, *y* , *u*2ð Þ *x*, *y* , … *un*ð Þ *x*, *y*

vector of unknown function and **<sup>F</sup>**ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>f</sup>* <sup>1</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *<sup>f</sup> <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>* is the

In addition, is took that two cases for the domain, the rectangular shape, and nonrectangular one and three cases relative to the geometry of the nonrectangular domain are considered where can be transformed into the rectangular shape [35]. The first one is <sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>θ</sup>*, *<sup>s</sup>* <sup>∈</sup> <sup>2</sup> : *<sup>a</sup>*<sup>≤</sup> *<sup>s</sup>*≤*b*, *<sup>g</sup>*1ð Þ*<sup>s</sup>* <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup> *<sup>g</sup>*2ð Þ*<sup>s</sup>* � � where *<sup>g</sup>*1ð Þ*<sup>s</sup>* and *g*2ð Þ*s* are continues functions of *s*, the second one can be consider as Ω ¼ ð Þ *<sup>θ</sup>*, *<sup>s</sup>* <sup>∈</sup> <sup>2</sup> : *<sup>c</sup>*<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>*d*, *<sup>g</sup>*1ð Þ*<sup>θ</sup>* <sup>≤</sup> *<sup>s</sup>*≤*g*2ð Þ*<sup>θ</sup>* � � where *<sup>g</sup>*1ð Þ*<sup>θ</sup>* and *<sup>g</sup>*2ð Þ*<sup>θ</sup>* are continues functions of *θ*, Also the last one is a domain which is neither of the first nor the second kinds but could be separated to finite numbers of first or second subdomains, it is labeled as a domain of third kind. Without loss of generality, the

*ς*¼0

≤ max *i*

<sup>≤</sup> <sup>X</sup>*<sup>n</sup> ς*¼0

<sup>∥</sup>*Et*∥∞ <sup>≤</sup> <sup>C</sup>*qK*ð Þ *<sup>λ</sup>*, *<sup>ς</sup> <sup>R</sup><sup>m</sup>*þ1�*μ*∥*u*ð Þ *<sup>m</sup>*þ<sup>1</sup>

In this section, we will provide an approximation solution of the 2-D linear system of Fredholm integral equations by the MLS method. The matrix form of this

<sup>∣</sup>*λς*∣*Rm*þ1�*<sup>ς</sup>* <sup>≤</sup> *<sup>K</sup>*ð Þ *<sup>λ</sup>*, *<sup>ς</sup> <sup>R</sup>m*þ1�*<sup>μ</sup>*

It should be noted that in the nonlinear system the upper bound of error depends

<sup>∣</sup>*λς*∣∥ *<sup>∂</sup><sup>ς</sup>*

X*n ς*¼0

∣*λς*k *∂ς*

C*q*∣*λς*∣∥*u*ð Þ *<sup>m</sup>*þ<sup>1</sup>

*<sup>t</sup><sup>ς</sup> U t*ðÞ� *<sup>∂</sup><sup>ς</sup>*

*<sup>t</sup><sup>ς</sup> <sup>U</sup><sup>h</sup>*

*<sup>t</sup><sup>ς</sup> ui*ð Þ�*<sup>t</sup>*

*<sup>k</sup>* <sup>∥</sup>*<sup>L</sup>*<sup>∞</sup> *<sup>R</sup>m*þ1�*<sup>ς</sup>*

*ς*¼0

*K x*ð Þ , *y*, *θ*, *s* **U**ð Þ *θ*, *s dθds*, ð Þ *x*, *y* ∈ Ω, (36)

, Also *K x*ð Þ¼ , *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup> <sup>κ</sup>ij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup>* � �, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup>

*<sup>k</sup>* ∥*<sup>L</sup>*<sup>∞</sup> *:*

∣*λς*∣, so demanded that

*<sup>T</sup>* is the

ð Þ*t* ∥∞

*∂ς tς uh <sup>i</sup>* ð Þ*t* ∣

$$\mathbf{K}(\mathbf{x}, \mathbf{y}, t, s) = \frac{\mathbf{g}\_2(\theta) - \mathbf{g}\_1(\theta)}{2} \mathbf{K}(\mathbf{x}, \mathbf{y}, \theta, s) \tag{40}$$

Also, the second kind is straight similarly by commuting the variables and the third kind can be separated to finite numbers of sub-domains of the first or second kinds, so the method can be applied in each sub-domain as described earlier.

So, for the numerical integration <sup>Ω</sup> <sup>¼</sup> <sup>⋃</sup>*<sup>L</sup> <sup>l</sup>*¼<sup>1</sup>Ω*<sup>l</sup>* and <sup>Ω</sup>*<sup>l</sup>* <sup>⋂</sup> <sup>Ω</sup>*<sup>k</sup>* 6¼ <sup>∅</sup>, 1≤*k*, *<sup>l</sup>* <sup>≤</sup>*<sup>L</sup>*

$$\underset{\Omega}{\text{g}}(\mathfrak{s})d\mathfrak{s} = \sum\_{l=1}^{L} \underset{\Omega\_{l}}{\text{g}}(\mathfrak{s})d\mathfrak{s} \tag{41}$$

Here, the MLS method is applied for the general case where the domain is ½ �� *a*, *b* ½ � *c*, *d* .

To apply the method, as described in section 2.1, instead of *ui* from *U*, we can replace *u<sup>h</sup> <sup>i</sup>* from (12). So we have

$$U^h(\mathbf{x}) = \begin{pmatrix} u\_1^h(\mathbf{x}), u\_2^h(\mathbf{x}), \dots, u\_n^h(\mathbf{x}) \end{pmatrix}^T \tag{42}$$

Also, obviously from (12)

$$u\_j^h(\mathbf{x}, \boldsymbol{\upchi}) = \sum\_{i=1}^N \phi\_i(\mathbf{x}, \boldsymbol{\upchi}) u\_j(\mathbf{x}\_i, \boldsymbol{\upchi}\_i) \tag{43}$$

in this section, is assumed that the domain has rectangular shape, so system (36) becomes as follows

$$\left(u\_1^h(\mathbf{x},\boldsymbol{\mathcal{y}}), u\_2^h(\mathbf{x},\boldsymbol{\mathcal{y}}), \dots \boldsymbol{u}\_n^h(\mathbf{x},\boldsymbol{\mathcal{y}})\right)^T = f(\mathbf{x},\boldsymbol{\mathcal{y}}) + \int\_c^d \left[\left[k\_{\vec{\boldsymbol{\eta}}}(\mathbf{x},\boldsymbol{\mathcal{y}},\mathbf{t},s)\right] \left(u\_1^h(\mathbf{t},s), u\_2^h(\mathbf{t},s), \dots u\_n^h(\mathbf{t},s)\right)^T \mathbf{d}t d\boldsymbol{s} \,. \tag{44}$$

By substituting (42) in (44), and it holds at points *xr*, *yr* � �,*<sup>r</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>* we have

Therefore any *uj*ð Þ *x*, *y* at any arbitrary point ð Þ *x*, *y* from the domain of the

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

*ϕi*ð Þ *x*, *y uji xi*, *yi*

� � (51)

Kð Þ *x*, *y*, *t*, *s* **U**ð Þ *t*, *s dtds*,ð Þ *x*, *y* ∈ Ω, (52)

*<sup>T</sup>*, *the  vector of unknown functions*

*<sup>i</sup>* from (12). So we have

*<sup>n</sup>*ð Þ **<sup>x</sup>** � �*<sup>T</sup>* (56)

<sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *uh*

<sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *<sup>u</sup><sup>h</sup>*

*a:* (54)

*c:* (55)

� � (57)

<sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *uh <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>*

<sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *uh <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>*

(53)

*dtds:*

(58)

*dθdξ:*

(59)

, *the  vector of known functions*

*N*

*i*¼1

very simple and effective. In this case, the domain under study is as Ω ¼

ð

Ω

*t x*ð Þ¼ , *<sup>θ</sup> <sup>x</sup>* � *<sup>a</sup>*

*s y*ð Þ¼ , *<sup>ξ</sup> <sup>y</sup>* � *<sup>c</sup>*

*b* � *a*

*d* � *c*

<sup>1</sup> ð Þ **<sup>x</sup>** , *uh*

*j*¼1

<sup>K</sup>ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup> <sup>κ</sup>ij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>* � �*i*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *n the matrix of kernels:*

like the Fredholm type, it is the matrix form of a system, so we have

Implementation of the proposed method on the Volterra integral equations is

½ �� *a*, *x* ½ � *c*, *y* such that 0 ≤*x*≤1, 0 ≤*y*≤ 1 and *a*,*c* are constant, so a Volterra system

By the following transformation the interval ½ � *a*, *x* and ½ � *c*, *y* can be transferred to

*θ* þ

*ξ* þ

*b* � *x b* � *a*

*d* � *y d* � *c*

<sup>2</sup>ð Þ **<sup>x</sup>** , … *<sup>u</sup><sup>h</sup>*

*ϕj*ð Þ **x** *ui* **x***<sup>j</sup>*

*<sup>κ</sup>ij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>* � �*: uh*

*<sup>κ</sup>ij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>* � �*: <sup>u</sup><sup>h</sup>*

*uj*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>≈</sup> <sup>X</sup>

problem, can be approximated by Eq. (43) as

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

type of integral equations can be consider as

**U**ð Þ¼ *x*, *y* ð Þ *u*1ð Þ *x*, *y* , *u*2ð Þ *x*, *y* , … *un*ð Þ *x*, *y*

**<sup>F</sup>**ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>f</sup>* <sup>1</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *<sup>f</sup>* <sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *<sup>f</sup> <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>*

Then instead of *ui* from *U*, we can replace *u<sup>h</sup>*

*<sup>U</sup><sup>h</sup>*ð Þ¼ **<sup>x</sup>** *<sup>u</sup><sup>h</sup>*

*uh*

where **x** ¼ ð Þ *x*, *y* ∈ ½ �� *a*, *b* ½ � *c*, *d* , thus, system (52) becomes

ð **x** ð **y**

*c*

Therefore from (54) and (55), the system (58) takes the following form

*a*

ð **b** ð **d**

*c*

*a*

*<sup>i</sup>* ð Þ¼ **<sup>x</sup>** <sup>X</sup> *N*

a fixed interval ½ � *a*, *b* and ½ � *c*, *d* ,

where

*uh* <sup>1</sup> ð Þ **<sup>x</sup>** , *uh*

> *uh* <sup>1</sup> ð Þ **<sup>x</sup>** , *uh*

**71**

<sup>2</sup>ð Þ **<sup>x</sup>** , … *uh <sup>n</sup>*ð Þ **<sup>x</sup>** � �*<sup>T</sup>* <sup>¼</sup> *<sup>F</sup>*ð Þþ **<sup>x</sup>**

<sup>2</sup>ð Þ **<sup>x</sup>** , … *<sup>u</sup><sup>h</sup> <sup>n</sup>*ð Þ **<sup>x</sup>** � �*<sup>T</sup>* <sup>¼</sup> *<sup>F</sup>*ð Þþ **<sup>x</sup>**

**U**ð Þ¼ *x*, *y* **F**ð Þþ *x*, *y*

**4.2 Volterra type**

$$\begin{split} \mathbf{f}\left(\mathbf{x}\_{r},\mathbf{y}\_{r}\right) &= \left(\sum\_{i=1}^{N} \phi\_{i}(\mathbf{x}\_{r},\mathbf{y}\_{r})u\_{1}(\mathbf{x}\_{i},\mathbf{y}\_{i}), \sum\_{i=1}^{N} \phi\_{i}(\mathbf{x}\_{r},\mathbf{y}\_{r})u\_{2}(\mathbf{x}\_{i},\mathbf{y}\_{i}), \dots \sum\_{i=1}^{N} \phi\_{i}(\mathbf{x}\_{r},\mathbf{y}\_{r})u\_{n}(\mathbf{x}\_{i},\mathbf{y}\_{i})\right)^{T} \\ &- \int\limits\_{\mathbf{c}} \int\limits\_{a} \left[k\_{\overline{\eta}}(\mathbf{x}\_{r},\mathbf{y}\_{r},\mathbf{t},\mathbf{s})\right] \left(\sum\_{i=1}^{N} \phi\_{i}(\mathbf{t},\mathbf{s})u\_{1}(\mathbf{x}\_{i},\mathbf{y}\_{i}), \dots, \sum\_{i=1}^{N} \phi\_{i}(\mathbf{t},\mathbf{s})u\_{n}(\mathbf{x}\_{i},\mathbf{y}\_{i})\right)^{T} \,d\mathbf{t}d\mathbf{s}. \end{split} \tag{45}$$

We consider the *m*1-point numerical integration scheme over Ω relative to the coefficients *τk*, *ς<sup>p</sup>* � � and weights *<sup>ω</sup><sup>k</sup>* and *<sup>ω</sup><sup>p</sup>* for solving integrals in (45), i.e.,

$$\phi(\mathbf{F}\_N)\_j \mu\_j(\mathbf{x}, y) = \sum\_{p=1}^{m\_1} \sum\_{k=1}^{m\_1} \left( \sum\_{i=1}^N k\_{ji}(\mathbf{x}\_r, y\_r, \tau\_k, \xi\_k) \phi\_i(\tau\_k, \xi\_k) a y\_k a\_p \right), (\mathbf{x}, y) \in \Omega, u\_i \in (-\infty, \infty) \tag{46}$$

Applying the numerical integration rule (46) yields

$$\begin{split} \mathbf{f}(\mathbf{x}\_{r},\mathbf{y}\_{r}) &= \left(\sum\_{i=1}^{N} \left(\phi\_{i}(\mathbf{x}\_{r},\mathbf{y}\_{r}) - \sum\_{p=1}^{m\_{1}} \sum\_{k=1}^{m\_{1}} \left(\sum\_{j=1}^{N} k\_{j1}(\mathbf{x}\_{r},\mathbf{y}\_{r},\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) \phi\_{i}(\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) a\_{k}a\_{l}ap\right)\right) \right) \mathbf{u}\_{1i}, \\ &\quad \sum\_{i=1}^{N} \left(\phi\_{i}(\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) - \sum\_{p=1}^{m\_{1}} \sum\_{k=1}^{m\_{1}} \left(\sum\_{j=1}^{N} k\_{j2}(\mathbf{x}\_{r},\mathbf{y}\_{r},\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) \phi\_{i}(\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) a\_{k}ap\right)\right) \mathbf{u}\_{2i}, \\ &\quad \dots \sum\_{i=1}^{N} \left(\phi\_{i}(\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) - \sum\_{p=1}^{m\_{1}} \sum\_{k=1}^{m\_{1}} \left(\sum\_{j=1}^{N} k\_{jn}(\mathbf{x}\_{r},\mathbf{y}\_{r},\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) \phi\_{i}(\mathbf{z}\_{k},\boldsymbol{\xi}\_{k}) a\_{k}ap\right)\right) \mathbf{u}\_{nl}\right)^{T}, r = 1,2,\dots,N. \end{split}$$

where *uj* � � *<sup>i</sup>* are the approximate quantities of *uj* when we use a quadrature rule instead of the exact integral. Now if we set *Fl*, *l* ¼ 1, 2, … *n* as a *N* by *N* matrices defined by:

$$\phi(F\_l)\_{i,j} = \phi\_i(\mathbf{x}\_r, \mathbf{y}\_r) - \sum\_{p=1}^{m\_1} \sum\_{k=1}^{m\_1} \left( \sum\_{j=1}^N k\_{jl} \left( \mathbf{x}\_r, \mathbf{y}\_r, \tau\_k, \mathbf{c}\_p \right) \phi\_i \left( \tau\_k, \mathbf{c}\_p \right) o\_k \right) o\_p \tag{47}$$

So, the moment matrix F is defined by (47) as follows

$$\mathbf{F} = [F\_1, F\_2, \dots \\ F\_n]\_{nN \times nN} \tag{48}$$

And

$$U = \left[ \left( u\_{11}, u\_{12}, \dots, u\_{1N} \right)^T, \left( u\_{21}, u\_{22}, \dots, u\_{2N} \right)^T, \dots \left( u\_{n1}, u\_{n2}, \dots, u\_{nN} \right)^T \right]^T$$

$$\mathbf{f} \left( \mathbf{x}\_r, \mathbf{y}\_r \right) = \left( \left[ \left( f\_1 \left( \mathbf{x}\_r, \mathbf{y}\_r \right) \right)\_{r=1}^N \right]^T, \left[ \left( f\_2 \left( \mathbf{x}\_r, \mathbf{y}\_r \right) \right)\_{r=1}^N \right]^T, \dots \left[ \left( f\_n \left( \mathbf{x}\_r, \mathbf{y}\_r \right) \right)\_{r=1}^N \right] \right)^T.$$

So by solving the following linear system of equations with a proper numerical method such as Gauss elimination method or etc. leads to quantities, *uji*.

$$\mathbf{F}U = \mathbf{f} \tag{50}$$

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

Therefore any *uj*ð Þ *x*, *y* at any arbitrary point ð Þ *x*, *y* from the domain of the problem, can be approximated by Eq. (43) as

$$u\_j(\mathbf{x}, \boldsymbol{y}) \approx \sum\_{i=1}^{N} \phi\_i(\mathbf{x}, \boldsymbol{y}) u\_{ji}(\mathbf{x}\_i, \boldsymbol{y}\_i) \tag{51}$$

#### **4.2 Volterra type**

By substituting (42) in (44), and it holds at points *xr*, *yr*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

� �,

X *N*

*ϕ<sup>i</sup> xr*, *yr*

*ϕi*ð Þ *t*, *s u*<sup>1</sup> *xi*, *yi*

We consider the *m*1-point numerical integration scheme over Ω relative to the

� �*ϕ<sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ*ωkω<sup>p</sup>* !

*kj*<sup>1</sup> *xr*, *yr*, *τk*, *ς<sup>k</sup>*

*kj*<sup>2</sup> *xr*, *yr*, *τk*, *ς<sup>k</sup>*

! !

instead of the exact integral. Now if we set *Fl*, *l* ¼ 1, 2, … *n* as a *N* by *N* matrices

*kjn xr*, *yr*, *τk*, *ς<sup>k</sup>*

*kjl xr*, *yr*, *τk*, *ς<sup>p</sup>* � �

*r*¼1

*<sup>i</sup>* are the approximate quantities of *uj* when we use a quadrature rule

!

! !

! !

� �*u*<sup>2</sup> *xi*, *yi*

� �, … ,

and weights *ω<sup>k</sup>* and *ω<sup>p</sup>* for solving integrals in (45), i.e.,

� �*ϕ<sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ*ωkω<sup>p</sup>*

� �*ϕ<sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ*ωkω<sup>p</sup>*

� �*ϕ<sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ*ωkω<sup>p</sup>*

*ϕ<sup>i</sup> τk*, *ς<sup>p</sup>* � �

<sup>F</sup> <sup>¼</sup> ½ � *<sup>F</sup>*1, *<sup>F</sup>*2, … *Fn nN*�*nN* (48)

*<sup>T</sup>*, … ð Þ *un*1, *un*2, … *unN*

, … *f <sup>n</sup> xr*, *yr* � � � � *<sup>N</sup>*

F*U* ¼ **f** (50)

� � !*<sup>T</sup>*

� �, … X

X *N*

� � !*<sup>T</sup>*

*i*¼1

*N*

*i*¼1

*i*¼1

*N*

*i*¼1

*kji xr*, *yr*, *τk*, *ς<sup>k</sup>*

X*N j*¼1

X*N j*¼1

> X*N j*¼1

X *N*

*j*¼1

*<sup>T</sup>*,ð Þ *<sup>u</sup>*21, *<sup>u</sup>*22, … *<sup>u</sup>*2*<sup>N</sup>*

, *f* <sup>2</sup> *xr*, *yr* � � � � *<sup>N</sup>*

method such as Gauss elimination method or etc. leads to quantities, *uji*.

*<sup>T</sup>* h i*<sup>T</sup>*

h i*<sup>T</sup>*

� � h i

So by solving the following linear system of equations with a proper numerical

have

**f** *xr*, *yr*

ð Þ Ϝ*<sup>N</sup> <sup>j</sup>*

**f** *xr*, *yr* � � <sup>¼</sup>

� � <sup>¼</sup> <sup>X</sup>

coefficients *τk*, *ς<sup>p</sup>*

*uj*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* <sup>X</sup>*<sup>m</sup>*<sup>1</sup>

 X*<sup>N</sup> i*¼1

> X*N i*¼1

… <sup>X</sup>*<sup>N</sup> i*¼1

ð Þ *Fl <sup>i</sup>*,*<sup>j</sup>* ¼ *ϕ<sup>i</sup> xr*, *yr*

*U* ¼ ð Þ *u*11, *u*12, … *u*1*<sup>N</sup>*

� � � � *<sup>N</sup>*

h i*<sup>T</sup>*

� � <sup>¼</sup> *<sup>f</sup>* <sup>1</sup> *xr*, *yr*

where *uj* � �

defined by:

And

**f** *xr*, *yr*

**70**

*N*

*i*¼1

ð *b*

*a*

� �

*p*¼1

*ϕ<sup>i</sup> xr*, *yr* � � �X*m*<sup>1</sup>

*<sup>ϕ</sup><sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ�X*m*<sup>1</sup>

*<sup>ϕ</sup><sup>i</sup> <sup>τ</sup>k*, *<sup>ς</sup><sup>k</sup>* ð Þ�X*m*<sup>1</sup>

� � �X*<sup>m</sup>*<sup>1</sup>

*p*¼1

X*m*1 *k*¼1

So, the moment matrix F is defined by (47) as follows

*r*¼1

X*m*1 *k*¼1

*c*

� ð *d* *ϕ<sup>i</sup> xr*, *yr*

� �*u*<sup>1</sup> *xi*, *yi*

*kij xr*, *yr*, *<sup>t</sup>*, *<sup>s</sup>* � � � � <sup>X</sup>

X *N*

*i*¼1

Applying the numerical integration rule (46) yields

X*m*1 *k*¼1

X*m*1 *k*¼1

> X*m*1 *k*¼1

*p*¼1

*p*¼1

*p*¼1

� �,*<sup>r</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>* we

*ϕ<sup>i</sup> xr*, *yr*

*ϕi*ð Þ *t*, *s un xi*, *yi*

� �*un xi*, *yi*

,ð Þ *x*, *y* ∈ Ω, *ui* ∈ ð Þ �∞, ∞

*u*1*i*,

*u*2*i*,

*ωk*

*r*¼1

*T :* (49)

*uni* ! *dtds:*

(45)

(46)

*<sup>T</sup>*,*<sup>r</sup>* <sup>¼</sup> 1, 2, … *<sup>N</sup>*

*ω<sup>p</sup>* (47)

Implementation of the proposed method on the Volterra integral equations is very simple and effective. In this case, the domain under study is as Ω ¼ ½ �� *a*, *x* ½ � *c*, *y* such that 0 ≤*x*≤1, 0 ≤*y*≤ 1 and *a*,*c* are constant, so a Volterra system type of integral equations can be consider as

$$\mathbf{U}(\mathbf{x},\boldsymbol{y}) = \mathbf{F}(\mathbf{x},\boldsymbol{y}) + \int\_{\Omega} \mathbf{K}(\mathbf{x},\boldsymbol{y},t,s)\mathbf{U}(t,s)dt ds,\\ (\mathbf{x},\boldsymbol{y}) \in \Omega,\tag{52}$$

like the Fredholm type, it is the matrix form of a system, so we have

$$\mathbf{U}(\mathbf{x},\mathbf{y}) = (u\_1(\mathbf{x},\mathbf{y}), u\_2(\mathbf{x},\mathbf{y}), \dots \\ u\_n(\mathbf{x},\mathbf{y}))^T, \text{ the vector of unknown functions}$$

$$\mathbf{F}(\mathbf{x},\mathbf{y}) = \begin{pmatrix} f\_1(\mathbf{x},\mathbf{y}), f\_2(\mathbf{x},\mathbf{y}), \dots \\ f\_1(\mathbf{x},\mathbf{y}), \dots \\ f\_0(\mathbf{x},\mathbf{y},t,\mathbf{s}) \end{pmatrix}^T, \text{ the vector of known functions} \tag{53}$$

$$\mathbf{K}(\mathbf{x},\mathbf{y},t,\mathbf{s}) = \begin{bmatrix} \kappa\_{\vec{\eta}\dagger}(\mathbf{x},\mathbf{y},t,\mathbf{s}) \end{bmatrix} \mathbf{i}, \mathbf{j} = \mathbf{1}, \mathbf{2}, \dots, n \text{ the matrix of kernels.}$$

By the following transformation the interval ½ � *a*, *x* and ½ � *c*, *y* can be transferred to a fixed interval ½ � *a*, *b* and ½ � *c*, *d* ,

$$t(\mathbf{x}, \theta) = \frac{\mathbf{x} - a}{b - a} \theta + \frac{b - \mathbf{x}}{b - a} a. \tag{54}$$

$$\sigma(\mathcal{y},\xi) = \frac{\mathcal{y}-c}{d-c}\xi + \frac{d-\mathcal{y}}{d-c}c.\tag{55}$$

Then instead of *ui* from *U*, we can replace *u<sup>h</sup> <sup>i</sup>* from (12). So we have

$$U^h(\mathbf{x}) = \begin{pmatrix} u\_1^h(\mathbf{x}), u\_2^h(\mathbf{x}), \dots \\ u\_1^h(\mathbf{x}), \dots \\ u\_2^h(\mathbf{x}) \end{pmatrix}^T \tag{56}$$

where

$$u\_i^h(\mathbf{x}) = \sum\_{j=1}^N \phi\_j(\mathbf{x}) u\_i(\mathbf{x}\_j) \tag{57}$$

where **x** ¼ ð Þ *x*, *y* ∈ ½ �� *a*, *b* ½ � *c*, *d* , thus, system (52) becomes

$$\left(\boldsymbol{u}\_{1}^{h}(\mathbf{x}), \boldsymbol{u}\_{2}^{h}(\mathbf{x}), \dots \boldsymbol{u}\_{n}^{h}(\mathbf{x})\right)^{T} = F(\mathbf{x}) + \int\_{a}^{\mathbf{x}} \left[\boldsymbol{\kappa}\_{\vec{\eta}}(\mathbf{x}, \boldsymbol{y}, t, \mathbf{s})\right] \cdot \left(\boldsymbol{u}\_{1}^{h}(\mathbf{x}, \boldsymbol{y}), \boldsymbol{u}\_{2}^{h}(\mathbf{x}, \boldsymbol{y}), \dots \boldsymbol{u}\_{n}^{h}(\mathbf{x}, \boldsymbol{y})\right)^{T} d\mathbf{t} d\boldsymbol{s}.\tag{58}$$

Therefore from (54) and (55), the system (58) takes the following form

$$\left(u\_1^h(\mathbf{x}), u\_2^h(\mathbf{x}), \dots \boldsymbol{u}\_n^h(\mathbf{x})\right)^T = F(\mathbf{x}) + \int\_a^b \left[\left[\kappa\_{\vec{\eta}}(\mathbf{x}, \boldsymbol{y}, t, s)\right], \left(u\_1^h(\mathbf{x}, \boldsymbol{y}), u\_2^h(\mathbf{x}, \boldsymbol{y}), \dots \boldsymbol{u}\_n^h(\mathbf{x}, \boldsymbol{y})\right)^T d\theta d\xi.\tag{59}$$

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Where

$$K(.,.,.,.) = \frac{\mathcal{X} - a}{b - a} \frac{\mathcal{Y} - c}{d - c} \mathbf{K}(.,.,.,.,.),\tag{60}$$

Applying the numerical integration rule (64) in (63) yields

yields the following approximate solution at any point ð Þ *x*, *t* ∈ Ω*:*

*uj*ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>≈</sup> <sup>X</sup>

ð *x* ð *y*

*c*

*a*

� � þX*<sup>m</sup>*<sup>1</sup>

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

Finding unknowns **U***<sup>h</sup>* by solving the nonlinear system of algebraic Eq. (65)

Two-dimensional nonlinear system of Volterra integral equations can be con-

where K, **F** are known function and **U** the vector of unknown functions are defined in (63) [27]. In order to apply the MLS approximation method, as same as the linear type, the interval ½ � *a*, *x* and ½ � *c*, *y* transferred to a fixed interval ½ � *a*, *b* and

*b* � *a*

Using the numerical integration technique (64) which applied in the Fredholm case yields the same final nonlinear system (65), so the approximation solution of **U**

In this section, the proposed method can be applied to the system of 2-dimensional linear and nonlinear integral equations [37] and the system of differential equations. Also, the results of the examples illustrate the effectiveness of the proposed method Also the relative errors for the collocation nodal points is used.

∥*uiex*ð Þ *x*, *y* ∥

<sup>∥</sup>*ei*∥∞ <sup>¼</sup> <sup>∥</sup>*uiex*ð Þ� *<sup>x</sup>*, *<sup>y</sup> uh*

*N*

*i*¼1

*p*¼1

*ϕi*ð Þ *x*, *y uji xi*, *yi*

*<sup>i</sup>* , *i* ¼ 1, 2, … , *n* instead of *ui* in *U* ¼ ð*u*1, *u*2, … , *un* from (12) is replaced.

ð *d* ð *b*

*a*

*K x*, *y*, *t*, *s*, **U***<sup>h</sup>*

*<sup>i</sup>* ð Þ *x*, *y* ∥

*c*

**y** � *c d* � *c* X*m*1 *k*¼1

*kij xr*, *yr*, *τk*, *ςp*, **U***<sup>h</sup> τk*, *ς<sup>p</sup>* h i � � � � *dtds*

� � (66)

*K x*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>*, **<sup>U</sup>***<sup>h</sup>*ð Þ *<sup>t</sup>*, *<sup>s</sup>* � �*dtds* (68)

ð Þ *<sup>t</sup>*, *<sup>s</sup>* � �, (69)

Kð Þ *x*, *y*, *θ*, *s*, **U**ð Þ *θ*, *s dθds*,ð Þ *x*, *y* ∈ Ω, (67)

(65)

*<sup>n</sup> xr*, *yr*

� � � � *<sup>T</sup>* <sup>¼</sup> *f xr*, *yr*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

*uh* <sup>1</sup> *xr*, *yr* � �, *uh*

<sup>2</sup> *xr*, *yr* � �, … , *uh*

**5.2 Volterra type**

½ � *<sup>c</sup>*, *<sup>d</sup>* . Then *uh*

*uh*

where

**6. Examples**

**73**

<sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *uh*

sidered as the following form

**U**ð Þ¼ *x*, *y* **F**ð Þþ *x*, *y*

So the nonlinear system (67) is converted to

<sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … , *uh <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>* <sup>¼</sup> *f x*ð Þþ , *<sup>y</sup>*

*K x*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>*, **<sup>U</sup>***<sup>h</sup>*ð Þ *<sup>t</sup>*, *<sup>s</sup>* � � <sup>¼</sup> **<sup>x</sup>** � *<sup>a</sup>*

would be found by solving this system of equations.

Using techniques employed in the Fredholm case yields the same final linear system where

$$\phi(F\_l)\_{i,j} = \phi\_i(\mathbf{x}\_r, \mathbf{y}\_r) - \sum\_{p=1}^{m\_1} \sum\_{k=1}^{m\_1} \left( \sum\_{j=1}^N k\_{jl} \left( \mathbf{x}\_r, \mathbf{y}\_r, \tau\_k, \mathbf{c}\_p \right) \phi\_i \left( \tau\_k, \mathbf{c}\_p \right) o\_k \right) o\_p \tag{61}$$

where *l* ¼ 1, 2, … , *n*.

#### **5. Nonlinear systems of two-dimensional integral equation**

#### **5.1 Fredholm type**

In the nonlinear system, the unknown function cannot be written as a linear combination of the unknown variables or functions that appear in them, so the matrix form of Fredholm integral equations defined as the following form [27].

$$\mathbf{U}(\mathbf{x},\boldsymbol{y}) = \mathbf{F}(\mathbf{x},\boldsymbol{y}) + \int\_{\Omega} \mathbf{K}(\mathbf{x},\boldsymbol{y},\theta,\mathbf{s},\mathbf{U}(\theta,\mathbf{s}))d\theta d\mathbf{s}, (\mathbf{x},\boldsymbol{y}) \in \Omega,\tag{62}$$

Where **U**ð Þ *x*, *y* , K and **F** are defined as,

$$\mathbf{U}(\mathbf{x},\mathbf{y}) = (u\_1(\mathbf{x},\mathbf{y}), u\_2(\mathbf{x},\mathbf{y}), \dots, u\_n(\mathbf{x},\mathbf{y}))^T$$

$$\mathbf{K}(\mathbf{x}, \mathbf{y}, \theta, s, \mathbf{U}(\theta, s)) = \begin{bmatrix} k\_{\vec{\cdot}\vec{\}}(\mathbf{x}, \mathbf{y}, \theta, s, \mathbf{U}(\theta, s)) \end{bmatrix}, i, j = 1, 2, \dots, n$$

$$\mathbf{F} = \begin{pmatrix} f\_1, f\_2, \dots, f\_n \end{pmatrix}^T$$

As mentioned above, we assume that Ω ¼ ½ �� *a*, *b* ½ � *c*, *d* .

To apply the aproximation MLS method, we estimate the unknown functions *ui* by (12), so the system (62) becomes the following form

$$\left(u\_1^h(\mathbf{x}, \boldsymbol{y}), u\_2^h(\mathbf{x}, \boldsymbol{y}), \dots u\_n^h(\mathbf{x}, \boldsymbol{y})\right)^T = f(\mathbf{x}, \boldsymbol{y}) + \int\_c^d \left[\left[k\_{\vec{\boldsymbol{\eta}}}(\mathbf{x}, \boldsymbol{y}, t, \boldsymbol{s}, \mathbf{U}^h(t, \boldsymbol{s}))\right] dt d\mathbf{s}\right] \tag{63}$$

We consider the *m*1-point numerical integration scheme over the domain under study relative to the coefficients *τk*, *ς<sup>p</sup>* � � and weights *<sup>ω</sup><sup>k</sup>* and *<sup>ω</sup><sup>p</sup>* for solving towdimentional integrals in (63), i.e.,

$$\mathbf{u}(\mathbb{F}\_N)\mu\_i(\mathbf{x},\boldsymbol{y}) = \sum\_{p=1}^{m\_1} \sum\_{k=1}^{m\_1} k\_{ji} \left(\mathbf{x}, \mathbf{y}, \tau\_k, \boldsymbol{\xi}\_p, \sum\_{j=1}^N \phi\_j \left(\tau\_k, \boldsymbol{\xi}\_p\right) u\_i(\mathbf{x}, \boldsymbol{y}) \alpha\_k \alpha\_p\right), (\mathbf{x}, \boldsymbol{y}) \in \Omega, u\_i \in (-\infty, \infty) \tag{64}$$

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

Applying the numerical integration rule (64) in (63) yields

$$\left(u\_1^h(\mathbf{x}\_r, \mathbf{y}\_r), u\_2^h(\mathbf{x}\_r, \mathbf{y}\_r), \dots, u\_n^h(\mathbf{x}\_r, \mathbf{y}\_r)\right)^T = f(\mathbf{x}\_r, \mathbf{y}\_r) + \sum\_{p=1}^{m\_1} \sum\_{k=1}^{m\_1} \left[k\_{\vec{\eta}}(\mathbf{x}\_r, \mathbf{y}\_r, \tau\_k, \mathbf{c}\_p, \mathbf{U}^h(\tau\_k, \mathbf{c}\_p))\right] \text{d}t \, d\mathbf{s} \tag{65}$$

Finding unknowns **U***<sup>h</sup>* by solving the nonlinear system of algebraic Eq. (65) yields the following approximate solution at any point ð Þ *x*, *t* ∈ Ω*:*

$$u\_{\vec{\jmath}}(\mathbf{x}, \boldsymbol{\jmath}) \approx \sum\_{i=1}^{N} \phi\_i(\mathbf{x}, \boldsymbol{\jmath}) u\_{\vec{\jmath}i}(\mathbf{x}\_i, \boldsymbol{\jmath}\_i) \tag{66}$$

#### **5.2 Volterra type**

Where

system where

ð Þ *Fl <sup>i</sup>*,*<sup>j</sup>* ¼ *ϕ<sup>i</sup> xr*, *yr*

where *l* ¼ 1, 2, … , *n*.

**5.1 Fredholm type**

form [27].

*uh*

ð Þ Ϝ*<sup>N</sup> <sup>i</sup>*

**72**

<sup>1</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* , *uh*

*ui*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>* <sup>X</sup>*<sup>m</sup>*<sup>1</sup>

� � �X*m*<sup>1</sup>

**U**ð Þ¼ *x*, *y* **F**ð Þþ *x*, *y*

Where **U**ð Þ *x*, *y* , K and **F** are defined as,

*p*¼1

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

X*m*1 *k*¼1

X *N*

*j*¼1

**5. Nonlinear systems of two-dimensional integral equation**

ð

Ω

In the nonlinear system, the unknown function cannot be written as a linear combination of the unknown variables or functions that appear in them, so the matrix form of Fredholm integral equations defined as the following

**U**ð Þ¼ *x*, *y* ð Þ *u*1ð Þ *x*, *y* , *u*2ð Þ *x*, *y* , … , *un*ð Þ *x*, *y*

**<sup>K</sup>**ð*x*, *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup>*, **<sup>U</sup>**ð Þ *<sup>θ</sup>*, *<sup>s</sup>* Þ ¼ *kij*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>θ</sup>*, *<sup>s</sup>*, **<sup>U</sup>**ð Þ *<sup>θ</sup>*, *<sup>s</sup>* � �, *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>*

To apply the aproximation MLS method, we estimate the unknown functions *ui*

ð *d* ð *b*

*a*

*ui*ð Þ *x*, *y ωkω<sup>p</sup>*

*c*

We consider the *m*1-point numerical integration scheme over the domain under

*ϕ<sup>j</sup> τk*, *ς<sup>p</sup>* � �

!

� �

X*N j*¼1

**F** ¼ *f* <sup>1</sup>, *f* <sup>2</sup>, … , *f <sup>n</sup>* � �*<sup>T</sup>*

As mentioned above, we assume that Ω ¼ ½ �� *a*, *b* ½ � *c*, *d* .

by (12), so the system (62) becomes the following form

*kji x*, *y*, *τk*, *ςp*,

<sup>2</sup>ð Þ *<sup>x</sup>*, *<sup>y</sup>* , … *uh <sup>n</sup>*ð Þ *<sup>x</sup>*, *<sup>y</sup>* � �*<sup>T</sup>* <sup>¼</sup> *f x*ð Þþ , *<sup>y</sup>*

study relative to the coefficients *τk*, *ς<sup>p</sup>*

X*m*1 *k*¼1

dimentional integrals in (63), i.e.,

*p*¼1

*<sup>K</sup>*ð Þ¼ *:*, *:*, *:*, *: <sup>x</sup>* � *<sup>a</sup>*

*b* � *a*

Using techniques employed in the Fredholm case yields the same final linear

*y* � *c d* � *c*

*kjl xr*, *yr*, *τk*, *ς<sup>p</sup>* � �

!

Kð Þ *:*, *:*, *:*, *:* , (60)

*ωk*

*ω<sup>p</sup>* (61)

*ϕ<sup>i</sup> τk*, *ς<sup>p</sup>* � �

**K**ð Þ *x*, *y*, *θ*, *s*, **U**ð Þ *θ*, *s dθds*,ð Þ *x*, *y* ∈ Ω, (62)

*T*

*kij <sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>*, **<sup>U</sup>***<sup>h</sup>*ð Þ *<sup>t</sup>*, *<sup>s</sup>* � � � � *dtds* (63)

,ð Þ *x*, *y* ∈ Ω, *ui* ∈ð Þ �∞, ∞

(64)

and weights *ω<sup>k</sup>* and *ω<sup>p</sup>* for solving tow-

Two-dimensional nonlinear system of Volterra integral equations can be considered as the following form

$$\mathbf{U}(\mathbf{x},\boldsymbol{\mathcal{y}}) = \mathbf{F}(\mathbf{x},\boldsymbol{\mathcal{y}}) + \int\_{\boldsymbol{\mathcal{z}}}^{\boldsymbol{\mathcal{x}}} \Big[ \mathbf{K}(\mathbf{x},\boldsymbol{\mathcal{y}},\boldsymbol{\theta},\boldsymbol{\varepsilon},\mathbf{U}(\boldsymbol{\theta},\boldsymbol{\varepsilon})) d\boldsymbol{\theta} d\boldsymbol{s}, (\boldsymbol{\mathcal{x}},\boldsymbol{\mathcal{y}}) \in \Omega,\tag{67}$$

where K, **F** are known function and **U** the vector of unknown functions are defined in (63) [27]. In order to apply the MLS approximation method, as same as the linear type, the interval ½ � *a*, *x* and ½ � *c*, *y* transferred to a fixed interval ½ � *a*, *b* and ½ � *<sup>c</sup>*, *<sup>d</sup>* . Then *uh <sup>i</sup>* , *i* ¼ 1, 2, … , *n* instead of *ui* in *U* ¼ ð*u*1, *u*2, … , *un* from (12) is replaced. So the nonlinear system (67) is converted to

$$\left(u\_1^h(\mathbf{x}, \boldsymbol{y}), u\_2^h(\mathbf{x}, \boldsymbol{y}), \dots, u\_n^h(\mathbf{x}, \boldsymbol{y})\right)^T = f(\mathbf{x}, \boldsymbol{y}) + \int\_{\boldsymbol{\varepsilon}}^d \Big[\overline{\mathcal{K}}(\mathbf{x}, \boldsymbol{y}, t, \boldsymbol{\varepsilon}, \mathbf{U}^h(t, \boldsymbol{\varepsilon})) dt ds \tag{68}$$

where

$$
\overline{K}(\mathbf{x}, \mathbf{y}, t, s, \mathbf{U}^h(t, s)) = \frac{\mathbf{x} - a}{b - a} \frac{\mathbf{y} - c}{d - c} K(\mathbf{x}, \mathbf{y}, t, s, \mathbf{U}^h(t, s)), \tag{69}
$$

Using the numerical integration technique (64) which applied in the Fredholm case yields the same final nonlinear system (65), so the approximation solution of **U** would be found by solving this system of equations.

#### **6. Examples**

In this section, the proposed method can be applied to the system of 2-dimensional linear and nonlinear integral equations [37] and the system of differential equations. Also, the results of the examples illustrate the effectiveness of the proposed method Also the relative errors for the collocation nodal points is used.

$$\|e\_i\|\_{\infty} = \frac{\|u\_{i\text{ex}}(\boldsymbol{\kappa}, \boldsymbol{y}) - u\_i^h(\boldsymbol{\kappa}, \boldsymbol{y})\|}{\|u\_{i\text{ex}}(\boldsymbol{\kappa}, \boldsymbol{y})\|}$$

where *uh <sup>i</sup>* is the approximate solution of the exact solution *uiex*. Linear and quadratic basis functions are utilized in computations.

#### **6.1 Example 1**

As the first example, we consider the following system of nonlinear Fredholm integral equations [27].

$$u\_1(\mathbf{x}, \mathbf{y}) = f\_1(\mathbf{x}, \mathbf{y}) + \int\_{\Omega} u\_1(s, t) u\_2(s, t) ds dt$$

$$u\_2(\mathbf{x}, \mathbf{y}) = f\_2(\mathbf{x}, \mathbf{y}) + \int\_{\Omega} u\_1(s, t) u\_2(s, t) + u\_2^2(s, t) ds dt$$

where Ω ¼ ½ �� 0, 1 ½ � 0, 1 *:* The exact solutions are *U x*ð Þ¼ , *y* ð Þ *x* þ *y*, *x* and the *F x*ð Þ¼ , *<sup>y</sup> <sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* � <sup>7</sup> 12, *<sup>x</sup>* � <sup>11</sup> <sup>12</sup> � �*:* The distribution of randomly nodes is shown in **Figure 1**. By attention to the irregular nodal points distribution, unsuitable *δ* can lead to a singular matrix A. So in this example, the adapted algorithm can tackle such problems. The MLS and MMLS shape functions are computed by using Algorithm 1, so the exact value of the radius of the domain of influence is not important; in fact, it is chosen as an initial value.

**Figure 2.**

**Table 1.**

**Table 2.**

**75**

*The condition numbers of a at a sample point p and δ* ¼ *0:05 for example 1. Using algorithm 1.*

**Sample points Cond(A) Result of algorithm**

**n** *x y initial final Newδ N.O.iteration* 1 0*:*2575 0*:*4733 1*:*1005*e* � 17 1*:*0117*e* � 06 0*:*1297 11 2 0*:*2575 0*:*6160 0 3*:*1111*e* � 06 0*:*1569 13 3 0*:*2575 0*:*9293 5*:*3204*e* � 17 5*:*2445*e* � 07 0*:*0974 8 4 0*:*2551 0*:*6160 0 3*:*1115*e* � 06 0*:*1569 13 5 0*:*6991 0*:*2435 2*:*7166*e* � 17 1*:*4652*e* � 07 0*:*0550 2

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

*Change of the radius and the condition number A at sample points (x,y) using algorithm 1, for example 1.*

**N** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** <sup>6</sup>*:*<sup>28</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>3</sup> <sup>3</sup>*:*<sup>45</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>15</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>4</sup> <sup>5</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>46</sup> � <sup>10</sup>�<sup>7</sup> <sup>8</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>7</sup> <sup>5</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>4</sup> <sup>5</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>4</sup> <sup>4</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>8</sup> <sup>1</sup>*:*<sup>34</sup> � <sup>10</sup>�<sup>7</sup> <sup>4</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup>*:*<sup>15</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>4</sup> <sup>5</sup>*:*<sup>14</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>12</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>46</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>7</sup> <sup>1</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>4</sup> <sup>5</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>4</sup> <sup>1</sup>*:*<sup>84</sup> � <sup>10</sup>�<sup>6</sup> <sup>6</sup>*:*<sup>64</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>34</sup> � <sup>10</sup>�<sup>6</sup> <sup>7</sup>*:*<sup>14</sup> � <sup>10</sup>�<sup>6</sup>

*Maximum relative errors for different points Gauss-Legendre quadrature rule δ* ¼ *2r*, *for example 1.*

**m=1** k k*e* **<sup>∞</sup> m=2** k k*e* **<sup>∞</sup> m=3** k k*e* **<sup>∞</sup>**

The condition numbers of the matrix A is shown in **Figure 2** and the determinant of A at sample points *p* is shown in **Figure 3**, where the radius of support domains for any nodal points is started from *δ* ¼ 0*:*05. Note that, there is a different radius of support domain for any node point, it might be increased due to the inappropriate distribution of scattered points by the algorithm. These variations are shown in **Table 1** for sample points ð Þ *x*, *y* , where *Cond A*ð Þ is the conditions number A, its initial case (*δ* ¼ 0*:*005) and final case (*Newδ*, ) and *N:O:iteration* is the number of iteration of the algorithm for determining a suitable radius of support domain.

In computing, *<sup>δ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>r</sup>* where *<sup>r</sup>* <sup>¼</sup> <sup>0</sup>*:*05 and *<sup>c</sup>* <sup>¼</sup> <sup>2</sup>ffiffi 3 <sup>p</sup> *r:* Also In MMLS, *w<sup>ν</sup>* ¼ 0*:*1, *ν* ¼ 1, 2, 3*:* It should be noted that, these values were also selected experimentally. Relative errors of the MLS method for different Gauss-Legendre number points at *m* ¼ 1, 2 and 3 compared in **Table 2**, also investigating the proposed methods shown that increasing the number of numerical integration points does not guarantee the error decreases. jkjk.

**Figure 1.** *The scatter data of example 1.*

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

**Figure 2.** *The condition numbers of a at a sample point p and δ* ¼ *0:05 for example 1. Using algorithm 1.*


**Table 1.**

where *uh*

**6.1 Example 1**

integral equations [27].

*F x*ð Þ¼ , *<sup>y</sup> <sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* � <sup>7</sup>

*<sup>i</sup>* is the approximate solution of the exact solution *uiex*. Linear and

As the first example, we consider the following system of nonlinear Fredholm

ð

*u*1ð Þ *s*, *t u*2ð Þ *s*, *t dsdt*

<sup>2</sup>ð Þ *s*, *t dsdt*

*<sup>u</sup>*1ð Þ *<sup>s</sup>*, *<sup>t</sup> <sup>u</sup>*2ð Þþ *<sup>s</sup>*, *<sup>t</sup> <sup>u</sup>*<sup>2</sup>

Ω

ð

Ω

where Ω ¼ ½ �� 0, 1 ½ � 0, 1 *:* The exact solutions are *U x*ð Þ¼ , *y* ð Þ *x* þ *y*, *x* and the

� �*:* The distribution of randomly nodes is shown in **Figure 1**. By attention to the irregular nodal points distribution, unsuitable *δ* can lead to a singular matrix A. So in this example, the adapted algorithm can tackle such problems. The MLS and MMLS shape functions are computed by using Algorithm 1, so the exact value of the radius of the domain of influence is not important;

The condition numbers of the matrix A is shown in **Figure 2** and the determinant of A at sample points *p* is shown in **Figure 3**, where the radius of support domains for any nodal points is started from *δ* ¼ 0*:*05. Note that, there is a different radius of support domain for any node point, it might be increased due to the inappropriate distribution of scattered points by the algorithm. These variations are shown in **Table 1** for sample points ð Þ *x*, *y* , where *Cond A*ð Þ is the conditions number A, its initial case (*δ* ¼ 0*:*005) and final case (*Newδ*, ) and *N:O:iteration* is the number of iteration

*ν* ¼ 1, 2, 3*:* It should be noted that, these values were also selected experimentally. Relative errors of the MLS method for different Gauss-Legendre number points at *m* ¼ 1, 2 and 3 compared in **Table 2**, also investigating the proposed methods shown that increasing the number of numerical integration points does not guar-

3

<sup>p</sup> *r:* Also In MMLS, *w<sup>ν</sup>* ¼ 0*:*1,

quadratic basis functions are utilized in computations.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*u*1ð Þ¼ *x*, *y f* <sup>1</sup>ð Þþ *x*, *y*

of the algorithm for determining a suitable radius of support domain.

In computing, *<sup>δ</sup>* <sup>¼</sup> <sup>2</sup>*<sup>r</sup>* where *<sup>r</sup>* <sup>¼</sup> <sup>0</sup>*:*05 and *<sup>c</sup>* <sup>¼</sup> <sup>2</sup>ffiffi

*u*2ð Þ¼ *x*, *y f* <sup>2</sup>ð Þþ *x*, *y*

12, *<sup>x</sup>* � <sup>11</sup> 12

in fact, it is chosen as an initial value.

antee the error decreases. jkjk.

**Figure 1.**

**74**

*The scatter data of example 1.*

*Change of the radius and the condition number A at sample points (x,y) using algorithm 1, for example 1.*


**Table 2.**

*Maximum relative errors for different points Gauss-Legendre quadrature rule δ* ¼ *2r*, *for example 1.*


**Table 3.**

*Compare relative errors and CPU times of MLS and MMLS for different points Gauss-Legendre quadrature rule,*ð Þ *m* ¼ *2* , *for example 1.*

**Figure 3.** *The determinant of a at a sample point p and δ* ¼ *0:05 for example 1. Using algorithm 1.*

In **Table 3**, we can see that the CPU times for solving the nonlinear system (65) are much larger in MMLS method; but, the errors are very smaller (**Figure 3**).

#### **6.2 Example 2**

Consider the system of linear Fredholm integral equations with [27].

$$K(\mathbf{x}, y, t, s) = \begin{pmatrix} \varkappa(t+s) & -t \\ t\mathbf{s} & (y+\varkappa)t \end{pmatrix},\tag{70}$$

**Table 4** shows relative errors and CPU times of MLS for different Gauss-Legendre number points at *m* ¼ 1, 3*:* As shown in **Table 5**, comparing the errors of MMLS and MLS method determines the capability and accuracy of the proposed technique to solve systems of linear Fredholm integral equations. This indicates the

*Compare relative errors and CPU times of MLS and MMLS for different points Gauss-Legendre quadrature*

**m=1 m=3**

**N** *u***<sup>1</sup>** *u***<sup>2</sup>** *CPU.T. u***<sup>1</sup>** *u***<sup>2</sup>** *CPU.T.* <sup>2</sup>*:*<sup>94</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>02</sup> � <sup>10</sup>�<sup>4</sup> <sup>108</sup>*:*<sup>957</sup> <sup>2</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>91</sup> � <sup>10</sup>�<sup>4</sup> <sup>152</sup>*:*<sup>1474</sup> <sup>1</sup>*:*<sup>79</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>89</sup> � <sup>10</sup>�<sup>4</sup> <sup>159</sup>*:*<sup>258</sup> <sup>1</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>37</sup> � <sup>10</sup>�<sup>4</sup> <sup>170</sup>*:*<sup>7879</sup> <sup>1</sup>*:*<sup>86</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>989</sup> � <sup>10</sup>�<sup>4</sup> <sup>198</sup>*:*<sup>135</sup> <sup>2</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>30</sup> � <sup>10</sup>�<sup>4</sup> <sup>252</sup>*:*<sup>75424</sup> <sup>1</sup>*:*<sup>86</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>986</sup> � <sup>10</sup>�<sup>4</sup> <sup>221</sup>*:*<sup>321</sup> <sup>2</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>29</sup> � <sup>10</sup>�<sup>4</sup> <sup>247</sup>*:*<sup>0093</sup> <sup>1</sup>*:*<sup>85</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>987</sup> � <sup>10</sup>�<sup>4</sup> <sup>308</sup>*:*<sup>987</sup> <sup>2</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>37</sup> � <sup>10</sup>�<sup>4</sup> <sup>314</sup>*:*<sup>8173</sup> <sup>1</sup>*:*<sup>85</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>980</sup> � <sup>10</sup>�<sup>4</sup> <sup>395</sup>*:*<sup>125</sup> <sup>1</sup>*:*<sup>87</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>86</sup> � <sup>10</sup>�<sup>4</sup> <sup>402</sup>*:*<sup>9594</sup>

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

*Relative errors and CPU times of MLS for different points Gauss-Legendre quadrature rule at m* ¼ *1*, *3*, *for*

**m N** *u*<sup>1</sup> *u*<sup>2</sup> *CPU.T. u*<sup>1</sup> *u*<sup>2</sup> *CPU.T.* 2 5 <sup>1</sup>*:*<sup>24</sup> � <sup>10</sup>�<sup>7</sup> <sup>3</sup>*:*<sup>89</sup> � <sup>10</sup>�<sup>6</sup> <sup>289</sup>*:*<sup>75</sup> <sup>4</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>7</sup> <sup>6</sup>*:*<sup>12</sup> � <sup>10</sup>�<sup>6</sup> <sup>297</sup>*:*<sup>7</sup>

 <sup>3</sup>*:*<sup>16</sup> � <sup>10</sup>�<sup>7</sup> <sup>5</sup>*:*<sup>23</sup> � <sup>10</sup>�<sup>7</sup> <sup>189</sup>*:*<sup>99</sup> <sup>2</sup>*:*<sup>16</sup> � <sup>10</sup>�<sup>8</sup> <sup>6</sup>*:*<sup>26</sup> � <sup>10</sup>�<sup>8</sup> <sup>210</sup>*:*<sup>09</sup> <sup>1</sup>*:*<sup>68</sup> � <sup>10</sup>�<sup>7</sup> <sup>4</sup>*:*<sup>13</sup> � <sup>10</sup>�<sup>7</sup> <sup>389</sup>*:*<sup>95</sup> <sup>2</sup>*:*<sup>02</sup> � <sup>10</sup>�<sup>8</sup> <sup>8</sup>*:*<sup>12</sup> � <sup>10</sup>�<sup>8</sup> <sup>390</sup>*:*<sup>15</sup> <sup>1</sup>*:*<sup>61</sup> � <sup>10</sup>�<sup>7</sup> <sup>3</sup>*:*<sup>88</sup> � <sup>10</sup>�<sup>7</sup> <sup>410</sup>*:*<sup>90</sup> <sup>2</sup>*:*<sup>04</sup> � <sup>10</sup>�<sup>8</sup> <sup>5</sup>*:*<sup>24</sup> � <sup>10</sup>�<sup>8</sup> <sup>425</sup>*:*<sup>87</sup> <sup>1</sup>*:*<sup>45</sup> � <sup>10</sup>�<sup>7</sup> <sup>3</sup>*:*<sup>80</sup> � <sup>10</sup>�<sup>7</sup> <sup>604</sup>*:*<sup>80</sup> <sup>1</sup>*:*<sup>98</sup> � <sup>10</sup>�<sup>8</sup> <sup>3</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>8</sup> <sup>618</sup>*:*<sup>44</sup> <sup>1</sup>*:*<sup>44</sup> � <sup>10</sup>�<sup>7</sup> <sup>3</sup>*:*<sup>84</sup> � <sup>10</sup>�<sup>7</sup> <sup>689</sup>*:*<sup>9574</sup> <sup>1</sup>*:*<sup>04</sup> � <sup>10</sup>�<sup>8</sup> <sup>6</sup>*:*<sup>87</sup> � <sup>10</sup>�<sup>8</sup> <sup>697</sup>*:*<sup>04</sup>

**MLS MMLS**

Comparing the errors of MMLS and MLS method determines the capability and accuracy of the proposed method to solve systems of linear Fredholm integral equations.

The third example that we want to approximate is the system of linear Volterra-

1 �1

, exp *<sup>x</sup>*�*<sup>y</sup>* ð Þ*:* It is important to note that

, (71)

!

*K x*ð Þ¼ , *<sup>y</sup>*, *<sup>t</sup>*, *<sup>s</sup>* ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* exp ð Þ *<sup>t</sup>*þ*<sup>s</sup>* ð Þ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* exp ð Þ *<sup>t</sup>*þ*<sup>s</sup>*

the linear transformation used in the experiment is only (55) and from (40) the

The domain is considered as <sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>2</sup> : <sup>0</sup>≤*x*≤1, 0 <sup>≤</sup>*y*<sup>≤</sup> ð Þ <sup>1</sup> � *<sup>y</sup>* � � so that

advantage of the proposed method over these systems of equations.

**6.3 Example 3**

**Table 4.**

*example 2.*

**Table 5.**

*rule at m* ¼ 2, *for example 2.*

kernel becomes

**77**

Fredholm integral equations with [27].

*<sup>y</sup>*∈½ � 0, 1 . Also the exact solutions are exp *<sup>x</sup>*þ*<sup>y</sup>*

Such that *U x*ð Þ¼ , *y* ð Þ *x* þ *y*, *x* is the vector of The exact solutions and the vector of unknown function is *F x*ð Þ¼ � , *<sup>y</sup>* <sup>1</sup> <sup>6</sup> ð Þþ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* <sup>1</sup> 3 , 4 <sup>3</sup> *<sup>x</sup>* � <sup>1</sup> <sup>3</sup> <sup>þ</sup> <sup>1</sup> <sup>3</sup> *<sup>y</sup> :* Also the domain of the problem determine by Ω ¼ ½ �� 0, 1 ½ � 0, 1 *:* In this example, initial value of *r* as radius of support domain set by 0*:*05. Also Algorithm 1 is used for producing shape function at *m* ¼ 1, 2, 3. In computing, we put *w<sup>ν</sup>* ¼ 0*:*1, *ν* ¼ 1, 2, 3*:*

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*


#### **Table 4.**

*Relative errors and CPU times of MLS for different points Gauss-Legendre quadrature rule at m* ¼ *1*, *3*, *for example 2.*


#### **Table 5.**

In **Table 3**, we can see that the CPU times for solving the nonlinear system (65)

**MMLS** k k*e* **<sup>∞</sup> MLS** k k*e* **<sup>∞</sup> CPU times**

**N** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup> MLS MMLS** <sup>3</sup>*:*<sup>78</sup> � <sup>10</sup>�<sup>10</sup> <sup>8</sup>*:*<sup>13</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>*:*<sup>46</sup> � <sup>10</sup>�<sup>7</sup> <sup>8</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>7</sup> <sup>389</sup>*:*<sup>9574</sup> <sup>2</sup>*:*<sup>9776</sup> � <sup>10</sup><sup>3</sup> <sup>1</sup>*:*<sup>35</sup> � <sup>10</sup>�<sup>10</sup> <sup>2</sup>*:*<sup>00</sup> � <sup>10</sup>�<sup>11</sup> <sup>4</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>8</sup> <sup>1</sup>*:*<sup>34</sup> � <sup>10</sup>�<sup>7</sup> <sup>410</sup>*:*<sup>9083</sup> <sup>2</sup>*:*<sup>1109</sup> � <sup>10</sup><sup>3</sup> <sup>8</sup>*:*<sup>63</sup> � <sup>10</sup>�<sup>11</sup> <sup>2</sup>*:*<sup>51</sup> � <sup>10</sup>�<sup>9</sup> <sup>6</sup>*:*<sup>12</sup> � <sup>10</sup>�<sup>7</sup> <sup>2</sup>*:*<sup>46</sup> � <sup>10</sup>�<sup>6</sup> <sup>634</sup>*:*<sup>8373</sup> <sup>3</sup>*:*<sup>2115</sup> � <sup>10</sup><sup>3</sup> <sup>3</sup>*:*<sup>99</sup> � <sup>10</sup>�<sup>10</sup> <sup>1</sup>*:*<sup>58</sup> � <sup>10</sup>�<sup>9</sup> <sup>1</sup>*:*<sup>84</sup> � <sup>10</sup>�<sup>6</sup> <sup>6</sup>*:*<sup>64</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>0331</sup> � <sup>10</sup><sup>3</sup> <sup>2</sup>*:*<sup>5844</sup> � <sup>10</sup><sup>3</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Compare relative errors and CPU times of MLS and MMLS for different points Gauss-Legendre quadrature*

*x t*ðÞ � þ *s t ts y*ð Þ þ *x t* 

3 , 4 <sup>3</sup> *<sup>x</sup>* � <sup>1</sup> <sup>3</sup> <sup>þ</sup> <sup>1</sup> <sup>3</sup> *<sup>y</sup> :* Also the domain of

Such that *U x*ð Þ¼ , *y* ð Þ *x* þ *y*, *x* is the vector of The exact solutions and the vector

<sup>6</sup> ð Þþ *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>* <sup>1</sup>

the problem determine by Ω ¼ ½ �� 0, 1 ½ � 0, 1 *:* In this example, initial value of *r* as radius of support domain set by 0*:*05. Also Algorithm 1 is used for producing shape

, (70)

are much larger in MMLS method; but, the errors are very smaller (**Figure 3**).

*The determinant of a at a sample point p and δ* ¼ *0:05 for example 1. Using algorithm 1.*

Consider the system of linear Fredholm integral equations with [27].

*K x*ð Þ¼ , *y*, *t*, *s*

function at *m* ¼ 1, 2, 3. In computing, we put *w<sup>ν</sup>* ¼ 0*:*1, *ν* ¼ 1, 2, 3*:*

of unknown function is *F x*ð Þ¼ � , *<sup>y</sup>* <sup>1</sup>

**6.2 Example 2**

**76**

**Figure 3.**

**Table 3.**

*rule,*ð Þ *m* ¼ *2* , *for example 1.*

*Compare relative errors and CPU times of MLS and MMLS for different points Gauss-Legendre quadrature rule at m* ¼ 2, *for example 2.*

**Table 4** shows relative errors and CPU times of MLS for different Gauss-Legendre number points at *m* ¼ 1, 3*:* As shown in **Table 5**, comparing the errors of MMLS and MLS method determines the capability and accuracy of the proposed technique to solve systems of linear Fredholm integral equations. This indicates the advantage of the proposed method over these systems of equations.

Comparing the errors of MMLS and MLS method determines the capability and accuracy of the proposed method to solve systems of linear Fredholm integral equations.

#### **6.3 Example 3**

The third example that we want to approximate is the system of linear Volterra-Fredholm integral equations with [27].

$$K(\mathbf{x}, y, t, s) = \begin{pmatrix} (\mathbf{x} + \mathbf{y}) \exp^{(t+s)} & (\mathbf{x} + \mathbf{y}) \exp^{(t+s)} \\ \mathbf{1} & -\mathbf{1} \end{pmatrix},\tag{71}$$

The domain is considered as <sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>2</sup> : <sup>0</sup>≤*x*≤1, 0 <sup>≤</sup>*y*<sup>≤</sup> ð Þ <sup>1</sup> � *<sup>y</sup>* � � so that *<sup>y</sup>*∈½ � 0, 1 . Also the exact solutions are exp *<sup>x</sup>*þ*<sup>y</sup>* , exp *<sup>x</sup>*�*<sup>y</sup>* ð Þ*:* It is important to note that the linear transformation used in the experiment is only (55) and from (40) the kernel becomes

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\mathbf{K}(.,...,.,.) = \frac{\mathcal{Y}-c}{d-c}\mathbf{K}(.,...,.,.). \tag{72}$$

In this example, the effect of increasing radius of the domain of influence *δ<sup>i</sup>* in MLS method on error has been investigated. Therefore the *δ<sup>i</sup>* was considered as follows

$$\delta\_i = \lambda r \; i = 1, 2, \dots, N \tag{73}$$

Then the relative error by MMLS shape function described in section (2.3) were obtained, using *δ* ¼ 7*r* and *w<sup>ν</sup>* ¼ 10, such that *ν* ¼ 1, 2, 3 as weights of additional coefficients for MMLS. We can see that in **Table 9** the errors of the system of linear Volterra-Fredholm integral equations are similar in both methods (i.e. MLS and

*Compare relative errors and CPU times of MLS and MMLS for w<sup>ν</sup>* ¼ *10 and 10 Gauss-Legendre points and*

**MLS MMLS CPU time**

**r** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup> MLS MMLS** *:*<sup>2</sup> <sup>1</sup>*:*<sup>09</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>19</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>52</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>4</sup> <sup>4</sup>*:*0556 3*:*<sup>3893</sup> *:*<sup>1</sup> <sup>1</sup>*:*<sup>24</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>5</sup> <sup>7</sup>*:*<sup>36</sup> � <sup>10</sup>�<sup>6</sup> <sup>38</sup>*:*0616 31*:*<sup>7945</sup> *:*<sup>05</sup> <sup>1</sup>*:*<sup>91</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>085</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>54</sup> � <sup>10</sup>�<sup>5</sup> <sup>8</sup>*:*<sup>77</sup> � <sup>10</sup>�<sup>6</sup> <sup>496</sup>*:*1746 409*:*<sup>0342</sup>

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

<sup>1</sup>ðÞ¼� *<sup>t</sup>* <sup>1002</sup>*u*1ðÞþ*<sup>t</sup>* <sup>1000</sup>*u*<sup>2</sup>

*u*1ðÞ¼ *t exp* ð Þ �2*t u*2ðÞ¼ *t exp* ð Þ �*t :*

In this numerical example, two scheme are compared and as explained the main task of the modified method tackle the singularity of the moment matrix. **Table 10** presents the maximum relative error by MLS on a set of evaluation points (with *h* ¼ 0*:*1 *and* 0*:*02) and *δ* ¼ 4*h and* 3*h*. Also in **Table 11** MLS and MMLS at different number of nodes for *h* ¼ 0*:*004 and *δ* ¼ 5*h and* 8*h*, were compared (**Tables 10–12**).

**r** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** 0.1 <sup>5</sup> � <sup>10</sup>�<sup>3</sup> <sup>4</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>4</sup> <sup>8</sup>*:*<sup>85</sup> � <sup>10</sup>�<sup>4</sup> <sup>2</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>3</sup> 0.02 <sup>5</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>2</sup> <sup>6</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>42</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>52</sup> � <sup>10</sup>�<sup>5</sup>

**Type** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** MLS <sup>1</sup>*:*<sup>03</sup> � <sup>10</sup><sup>0</sup> <sup>0</sup>*:*<sup>98</sup> � 101 <sup>1</sup>*:*<sup>01</sup> � <sup>10</sup><sup>0</sup> <sup>9</sup>*:*<sup>2</sup> � 100 MMLS <sup>9</sup>*:*<sup>23</sup> � <sup>10</sup>�<sup>4</sup> <sup>9</sup>*:*<sup>22</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>89</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>96</sup> � <sup>10</sup>�<sup>4</sup>

*Maximum relative errors for h = 0.004 by MMLS and MLS, example 4.*

**m = 2,***δ* **=4r m = 2,** *δ* **=3r**

**m = 3,** *δ* **=5r m = 3,** *δ* **=8r**

<sup>2</sup>ðÞ¼ *<sup>t</sup> <sup>u</sup>*1ðÞ�*<sup>t</sup> <sup>u</sup>*2ðÞ�*<sup>t</sup> <sup>u</sup>*<sup>2</sup>

With the initial condition *u*1ð Þ¼ 0 1 and *u*2ð Þ¼ 0 1*:* The exact solution is

<sup>2</sup>ð Þ*t*

<sup>2</sup>ð Þ*t*

Consider the following nonlinear stiff systems of ODEs [38].

*u*0

(

*different values of δ* ¼ *7r at m* ¼ *2 for example 3.*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

*u*0

MMLS methods).

**Table 9.**

**6.4 Example 4**

**Table 10.**

**Table 11.**

**79**

*Maximum relative errors by MLS, example 4.*

In this way, useful information is obtained about the performance of the proposed method. By investigating the results in **Tables 6** and **7** we found that the relative error in MLS was also related to the radius of the domain of influence (i.e. *δ* ¼ *λr* so that *λ* ¼ 3, 5, 7); however, it cannot be greater than 7*:* For example, the relative errors by choosing *λ* ¼ 10 (i.e. *δ* ¼ 0*:*05*λ*) and 10�point GaussLegendre quadrature rule are k k*<sup>e</sup>* <sup>∞</sup>*u*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>2</sup> and k k*<sup>e</sup>* <sup>∞</sup>*u*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>2</sup> at *<sup>m</sup>* <sup>¼</sup> <sup>1</sup>*:* Also, **Table 8** depicts, the number of points in the numerical integration rule cannot be effective to increase the accuracy of the method.


**Table 6.**

*Maximum relative errors of MLS for 10 Gauss-Legendre points and different values of δ* ¼ *λr at m* ¼ *2*, *for example 3.*


**Table 7.**

*Maximum relative errors of MLS for 10 Gauss-Legendre points and different values of δ* ¼ *λr at m* ¼ 3 *for example 3.*


**Table 8.**

*Maximum relative errors of MLS for different values of δ* ¼ *λr*, *r* ¼ *0:05 and Gauss-Legendre points at m* ¼ *1*, *using algorithm 1 for example 3.*

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*


#### **Table 9.**

*<sup>K</sup>*ð Þ¼ *:*, *:*, *:*, *: <sup>y</sup>* � *<sup>c</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

effective to increase the accuracy of the method.

follows

**Table 6.**

*example 3.*

**Table 7.**

*example 3.*

**Table 8.**

**78**

*m* ¼ *1*, *using algorithm 1 for example 3.*

*d* � *c*

In this example, the effect of increasing radius of the domain of influence *δ<sup>i</sup>* in MLS method on error has been investigated. Therefore the *δ<sup>i</sup>* was considered as

In this way, useful information is obtained about the performance of the proposed method. By investigating the results in **Tables 6** and **7** we found that the relative error in MLS was also related to the radius of the domain of influence (i.e. *δ* ¼ *λr* so that *λ* ¼ 3, 5, 7); however, it cannot be greater than 7*:* For example, the relative errors by choosing *λ* ¼ 10 (i.e. *δ* ¼ 0*:*05*λ*) and 10�point GaussLegendre quadrature rule are k k*<sup>e</sup>* <sup>∞</sup>*u*<sup>1</sup> <sup>¼</sup> <sup>3</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>2</sup> and k k*<sup>e</sup>* <sup>∞</sup>*u*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>2</sup> at *<sup>m</sup>* <sup>¼</sup> <sup>1</sup>*:* Also, **Table 8** depicts, the number of points in the numerical integration rule cannot be

*λ* ¼ **3**k k*e* **<sup>∞</sup>** *λ* ¼ **5**k k*e* **<sup>∞</sup>** *λ* ¼ **7**k k*e* **<sup>∞</sup>**

**r** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *:*<sup>2</sup> <sup>9</sup> � <sup>10</sup>�<sup>3</sup> <sup>5</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>6</sup>*:*<sup>02</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>36</sup> � <sup>10</sup>�<sup>4</sup> *:*<sup>1</sup> <sup>1</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>3</sup> <sup>6</sup>*:*<sup>87</sup> � <sup>10</sup>�<sup>4</sup> <sup>1</sup>*:*<sup>34</sup> � <sup>10</sup>�<sup>4</sup> <sup>7</sup>*:*<sup>46</sup> � <sup>10</sup>�<sup>5</sup> <sup>4</sup>*:*<sup>57</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>91</sup> � <sup>10</sup>�<sup>6</sup> *:*<sup>05</sup> <sup>1</sup>*:*<sup>44</sup> � <sup>10</sup>�<sup>4</sup> <sup>8</sup>*:*<sup>14</sup> � <sup>10</sup>�<sup>5</sup> <sup>2</sup>*:*<sup>23</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>26</sup> � <sup>10</sup>�<sup>5</sup> <sup>6</sup>*:*<sup>27</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>6</sup> � <sup>10</sup>�<sup>6</sup>

*Maximum relative errors of MLS for 10 Gauss-Legendre points and different values of δ* ¼ *λr at m* ¼ *2*, *for*

**r** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *:*<sup>2</sup> <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>3</sup> <sup>8</sup>*:*<sup>77</sup> � <sup>10</sup>�<sup>3</sup> <sup>6</sup>*:*<sup>56</sup> � <sup>10</sup>�<sup>5</sup> <sup>3</sup>*:*<sup>01</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>09</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>19</sup> � <sup>10</sup>�<sup>5</sup> *:*<sup>1</sup> <sup>1</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>51</sup> � <sup>10</sup>�<sup>5</sup> <sup>3</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>5</sup> <sup>2</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>24</sup> � <sup>10</sup>�<sup>5</sup> <sup>7</sup>*:*<sup>22</sup> � <sup>10</sup>�<sup>6</sup> *:*<sup>05</sup> <sup>6</sup>*:*<sup>63</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>93</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>31</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>41</sup> � <sup>10</sup>�<sup>6</sup> <sup>1</sup>*:*<sup>38</sup> � <sup>10</sup>�<sup>6</sup>

*Maximum relative errors of MLS for 10 Gauss-Legendre points and different values of δ* ¼ *λr at m* ¼ 3 *for*

**N** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** *u***<sup>1</sup>** *u***<sup>2</sup>** <sup>2</sup>*:*<sup>2</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>3</sup> <sup>6</sup>*:*<sup>33</sup> � <sup>10</sup>�<sup>5</sup> <sup>4</sup>*:*<sup>52</sup> � <sup>10</sup>�<sup>5</sup> <sup>2</sup>*:*<sup>21</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>59</sup> � <sup>10</sup>�<sup>5</sup> <sup>1</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>7</sup>*:*<sup>4</sup> � <sup>10</sup>�<sup>4</sup> <sup>4</sup>*:*<sup>17</sup> � <sup>10</sup>�<sup>5</sup> <sup>2</sup>*:*<sup>89</sup> � <sup>10</sup>�<sup>5</sup> <sup>2</sup>*:*<sup>96</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>72</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>3</sup> <sup>7</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>07</sup> � <sup>10</sup>�<sup>5</sup> <sup>6</sup>*:*<sup>31</sup> � <sup>10</sup>�<sup>6</sup> <sup>5</sup>*:*<sup>11</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>3</sup> <sup>8</sup>*:*<sup>06</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>17</sup> � <sup>10</sup>�<sup>5</sup> <sup>3</sup>*:*<sup>96</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>69</sup> � <sup>10</sup>�<sup>6</sup> <sup>2</sup>*:*<sup>1</sup> � <sup>10</sup>�<sup>3</sup> <sup>1</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>3</sup> <sup>8</sup>*:*<sup>16</sup> � <sup>10</sup>�<sup>5</sup> <sup>5</sup>*:*<sup>17</sup> � <sup>10</sup>�<sup>5</sup> <sup>4</sup>*:*<sup>11</sup> � <sup>10</sup>�<sup>6</sup> <sup>3</sup>*:*<sup>74</sup> � <sup>10</sup>�<sup>6</sup>

*Maximum relative errors of MLS for different values of δ* ¼ *λr*, *r* ¼ *0:05 and Gauss-Legendre points at*

*λ* ¼ **3**k k*e* **<sup>∞</sup>** *λ* ¼ **5**k k*e* **<sup>∞</sup>** *λ* ¼ **7**k k*e* **<sup>∞</sup>**

*λ* ¼ **3**k k*e* **<sup>∞</sup>** *λ* ¼ **5**k k*e* **<sup>∞</sup>** *λ* ¼ **7**k k*e* **<sup>∞</sup>**

Kð Þ *:*, *:*, *:*, *: :* (72)

*δ<sup>i</sup>* ¼ *λr i* ¼ 1, 2, … , *N* (73)

*Compare relative errors and CPU times of MLS and MMLS for w<sup>ν</sup>* ¼ *10 and 10 Gauss-Legendre points and different values of δ* ¼ *7r at m* ¼ *2 for example 3.*

Then the relative error by MMLS shape function described in section (2.3) were obtained, using *δ* ¼ 7*r* and *w<sup>ν</sup>* ¼ 10, such that *ν* ¼ 1, 2, 3 as weights of additional coefficients for MMLS. We can see that in **Table 9** the errors of the system of linear Volterra-Fredholm integral equations are similar in both methods (i.e. MLS and MMLS methods).

#### **6.4 Example 4**

Consider the following nonlinear stiff systems of ODEs [38].

$$\begin{cases} \boldsymbol{u}\_1'(t) = -\mathbf{1}\,002\boldsymbol{u}\_1(t) + \mathbf{1}\,000\boldsymbol{u}\_2^2(t) \\ \boldsymbol{u}\_2'(t) = \boldsymbol{u}\_1(t) - \boldsymbol{u}\_2(t) - \boldsymbol{u}\_2^2(t) \end{cases}$$

With the initial condition *u*1ð Þ¼ 0 1 and *u*2ð Þ¼ 0 1*:* The exact solution is

$$\begin{aligned} u\_1(t) &= \exp\left(-2t\right), \\ u\_2(t) &= \exp\left(-t\right). \end{aligned}$$

In this numerical example, two scheme are compared and as explained the main task of the modified method tackle the singularity of the moment matrix. **Table 10** presents the maximum relative error by MLS on a set of evaluation points (with *h* ¼ 0*:*1 *and* 0*:*02) and *δ* ¼ 4*h and* 3*h*. Also in **Table 11** MLS and MMLS at different number of nodes for *h* ¼ 0*:*004 and *δ* ¼ 5*h and* 8*h*, were compared (**Tables 10–12**).


#### **Table 10.**

*Maximum relative errors by MLS, example 4.*


#### **Table 11.**

*Maximum relative errors for h = 0.004 by MMLS and MLS, example 4.*


**Table 12.**

*Maximum relative errors by MLS t*∈½ � *0*, *5 ,h* ¼ *0:004,, example 2.*

#### **6.5 Example 5**

In this example, we consider *U t*ðÞ¼ <sup>1</sup> <sup>47</sup> 95 exp ð Þ �2*<sup>t</sup>* � <sup>48</sup> *exp* ð Þ �96*<sup>t</sup>* , <sup>1</sup> <sup>47</sup> ð48 exp ð Þ� �96*t* exp ð Þ �2*t* ÞÞ as the exact solution and *U*ð Þ¼ 0, 0 ð Þ 1, 1 as the initial conditions for the following system of ODE,

$$\begin{cases} \varkappa'(t) = -\varkappa(t) + \mathfrak{B}\mathfrak{y}(t) \\ \jmath'(t) = -\varkappa(t) - \mathfrak{B}\mathfrak{y}(t) \end{cases}$$

**Table 12** presents the maximum relative norm of the errors on a fine set of evaluation points (with *h* ¼ 0*:*004) and *δ* ¼ 5*h* for MLS and MMLS at different type of weight functions. As seen in this table, one major advantage of MMLS is that the computational time used by MMLS is less than MLS.

#### **7. Conclusion**

In this paper, two meshless techniques called moving least squares and modified Moving least-squares approximation are applied for solving the system of functional equations. Comparing the results obtained by these methods with the results obtained by the exact solution shows that the moving least squares methods are the reliable and accurate methods for solving a system of functional equations. Meshless methods are free from choosing the domain and this makes it suitable to study realworld problems. Also, the modified algorithm has changed the ability to select the support range radius In fact, the user can begin to solve any problem with an arbitrary radius from the domain and the proposed algorithm can correct it during execution.

**Author details**

Mazandaran, Iran

**81**

Massoumeh Poura'bd Rokn Saraei\* and Mashaallah Matinfar

provided the original work is properly cited.

Department of Mathematics, Science of Mathematics Faculty, University of

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems…*

*DOI: http://dx.doi.org/10.5772/intechopen.89394*

\*Address all correspondence to: m.pourabd@gmail.com and m.matinfar@umz.ac.ir

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems… DOI: http://dx.doi.org/10.5772/intechopen.89394*

### **Author details**

**6.5 Example 5**

**Table 12.**

**7. Conclusion**

execution.

**80**

In this example, we consider *U t*ðÞ¼ <sup>1</sup>

*Maximum relative errors by MLS t*∈½ � *0*, *5 ,h* ¼ *0:004,, example 2.*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

conditions for the following system of ODE,

<sup>47</sup> 95 exp ð Þ �2*<sup>t</sup>* � <sup>48</sup> *exp* ð Þ �96*<sup>t</sup>* , <sup>1</sup>

 ð48 exp ð Þ� �96*t* exp ð Þ �2*t* ÞÞ as the exact solution and *U*ð Þ¼ 0, 0 ð Þ 1, 1 as the initial

**weight type** *u***<sup>1</sup>** *u***<sup>2</sup> Cputime** *u***<sup>1</sup>** *u***<sup>2</sup> Cputime** Guass <sup>3</sup>*:*<sup>06</sup> � <sup>10</sup>�<sup>3</sup> <sup>9</sup>*:*<sup>92</sup> � <sup>10</sup>�<sup>5</sup> <sup>61</sup>*:*<sup>1706</sup> <sup>8</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>4</sup> <sup>6</sup>*:*<sup>5</sup> � <sup>10</sup>�<sup>4</sup> <sup>0</sup>*:*<sup>5598</sup> Spline <sup>5</sup>*:*<sup>06</sup> � <sup>10</sup>�<sup>4</sup> <sup>5</sup>*:*<sup>3</sup> � <sup>10</sup>�<sup>4</sup> <sup>64</sup>*:*<sup>5897</sup> <sup>1</sup>*:*<sup>93</sup> � <sup>10</sup>�<sup>2</sup> <sup>4</sup>*:*<sup>23</sup> � <sup>10</sup>�<sup>4</sup> <sup>0</sup>*:*<sup>6714</sup> RBF <sup>6</sup>*:*<sup>407</sup> � <sup>10</sup>�<sup>4</sup> <sup>3</sup>*:*<sup>02</sup> � <sup>10</sup>�<sup>4</sup> <sup>59</sup>*:*<sup>1790</sup> <sup>6</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>3</sup> <sup>6</sup>*:*<sup>9</sup> � <sup>10</sup>�<sup>3</sup> <sup>0</sup>*:*<sup>5768</sup>

ðÞ¼� *t x t*ðÞþ 95*y t*ð Þ

**MLS MMLS**

ðÞ¼� *t x t*ðÞ� 97*y t*ð Þ

In this paper, two meshless techniques called moving least squares and modified Moving least-squares approximation are applied for solving the system of functional equations. Comparing the results obtained by these methods with the results obtained by the exact solution shows that the moving least squares methods are the reliable and accurate methods for solving a system of functional equations. Meshless methods are free from choosing the domain and this makes it suitable to study realworld problems. Also, the modified algorithm has changed the ability to select the support range radius In fact, the user can begin to solve any problem with an arbitrary radius from the domain and the proposed algorithm can correct it during

**Table 12** presents the maximum relative norm of the errors on a fine set of evaluation points (with *h* ¼ 0*:*004) and *δ* ¼ 5*h* for MLS and MMLS at different type of weight functions. As seen in this table, one major advantage of MMLS is that the

*x*0

computational time used by MMLS is less than MLS.

*y*0

47

Massoumeh Poura'bd Rokn Saraei\* and Mashaallah Matinfar Department of Mathematics, Science of Mathematics Faculty, University of Mazandaran, Iran

\*Address all correspondence to: m.pourabd@gmail.com and m.matinfar@umz.ac.ir

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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integral equations of first and second kinds. Zeitschrift für Naturforschung. 2008;**63a**(10):752-756

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**39**:484-493

2009

113-121

[1] Scudo FM. Vito Volterra and theoretical ecology. Theoretical Population Biology. 1971;**2**:1-23

[3] TeBeest KG. Numerical and analytical solutions of Volterra's population model. SIAM Review. 1997;

[4] Wazwaz AM. Partial Differential Equations and Solitary Waves Theory. Beijing and Berlin: HEP and Springer;

[5] Mckee S, Tang T, Diogo T. An Eulertype method for two-dimensional Volterra integral equations of the first kind. IMA Journal of Numerical Analysis. 2000;**20**:423-440

[6] Hanson R, Phillips J. Numerical solution of two-dimensional integral equations using linear elements. SIAM Journal on Numerical Analysis. 1978;**15**:

[7] Babolian E, Masouri Z. Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions. Journal of Computational and Applied Mathematics. 2008;**220**:51-57

[8] Beltyukov BA, Kuznechichina LN. A RungKutta method for the solution of two-dimensional nonlinear Volterra integral equations. Differential Equations. 1976;**12**:1169-1173

[9] Masouri Z, Hatamzadeh-Varmazyar S, Babolian E. Numerical method for solving system of Fredholm integral equations using Chebyshev cardinal functions. Advanced Computational Techniques in

Electromagnetics. 2014:1-13

**82**

[2] Small RD. Population growth in a closed model. In: Mathematical

Modelling: Classroom Notes in Applied Mathematics. Philadelphia: SIAM; 1989

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

[10] Jafarian A, Nia SAM, Golmankhandh AK, Baleanu D. Numerical solution of linear integral equations system using the Bernstein collocation method. Advances in Difference Equations. 2013:1-23

[11] Singh P. A note on the solution of two-dimensional Volterra integral equations by spline. Indian Journal of

[12] Assari P, Adibi H, Dehghan M. A meshless method based on the moving least squares (MLS) approximation for

Mathematics. 1979;**18**:61-64

the numerical solution of twodimensional nonlinear integral equations of the second kind on nonrectangular domains. Numerical Algorithm. 2014;**67**:423-455

[13] Brunner H, Kauthen JP. The numerical solution of two-dimensional

[14] Wazwaz A. Linear and Nonlinear Integral Equations Methods and

[15] Dehghan M, Abbaszadeha M, Mohebbib A. The numerical solution of the two-dimensional sinh-Gordon equation via three meshless methods. Engineering Analysis with Boundary

[16] Mirzaei D, Schaback R, Dehghan M. On generalized moving least squares and diffuse derivatives. IMA Journal of Numerical Analysis. 2012;**32**:983-1000

[17] Salehi R, Dehghan M. A generalized moving least square reproducing kernel method. Journal of Computational and Applied Mathematics. 2013;**249**:120-132

[18] Saadatmandi A, Dehghan M. A collocation method for solving Abels

Elements. 2015;**51**:220-235

Volterra integral equations by collocation and iterated collocation. IMA Journal of Numerical Analysis.

1989;**9**:47-59

Applications. 2011

[19] Dehghan M, Mirzaei D. Numerical solution to the unsteady twodimensional Schrodinger equation using meshless local boundary integral equation method. International Journal for Numerical Methods in Engineering. 2008;**76**:501-520

[20] Mukherjee YX, Mukherjee S. The boundary node method for potential problems. International Journal for Numerical Methods in Engineering. 1997;**40**:797-815

[21] Salehi R, Dehghan M. A moving least square reproducing polynomial meshless method. Applied Numerical Mathematics. 2013;**69**:34-58

[22] Matin far M, Pourabd M. Moving least square for systems of integral equations. Applied Mathematics and Computation. 2015;**270**:879-889

[23] Li S, Liu WK. Meshfree Particle Methods. Berlin: Springer-Verlag; 2004

[24] Liu GR, Gu YT. A matrix triangularization algorithm for the polynomial point interpolation method. Computer Methods in Applied Mechanics and Engineering. 2003;**192**: 2269-2295

[25] Belinha J. Meshless Methods in Biomechanics. Springer; 2014

[26] Chen S. Building interpolating and approximating implicit surfaces using moving least squares [Phd thesis] EECS-2007-14. Berkeley: EECS Department, University of California. 2007

[27] Matin far M, Pourabd M. Modified moving least squares method for twodimensional linear and nonlinear systems of integral equations. Computational and Applied Mathematics. 2018;**37**:5857-5875

[28] Joldesa GR, Chowdhurya HA, Witteka A, Doylea B, Miller K. Modified moving least squares with polynomial bases for scattered data approximation. Applied Mathematics and Computation. 2015;**266**:893-902

[29] Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Mathematics of Computation. 1981;**37**:141-158

[30] Wendland H. Local polynomial reproduction, and moving least squares approximation. IMA Journal of Numerical Analysis. 2001;**21**:285-300

[31] Zuppa C. Error estimates for moving least square approximations. Bulletin of the Brazilian Mathematical Society, New Series. 2001;**34**:231249

[32] Liu GR. Mesh Free Methods: Moving beyond the Finite Element Method. Boca Raton: CRC Press; 2003

[33] Zuppa C. Error estimates for moving least squareapproximations. Bulletin of the Brazilian Mathematical Society, New Series. 2003;**34**(2):231-249

[34] Zuppa C. Good quality point sets error estimates for moving least square approximations. Applied Numerical Mathematics. 2003;**47**:575-585

[35] Mirzaei D, Dehghan M. A meshless based method for solution of integral equations. Applied Numerical Mathematics. 2010;**60**:245-262

[36] McLain D. Two dimensional interpolation from random data. Computer Journal. 1976;**19**:178181

[37] Pourabd M. Meshless method based moving least squares for solving systems of integral equations with investigating the computational complexity of algorithms [Phd thesis], IRANDOC-2408208. IRAN: Department of mathematic, University of Mazandaran. 2017

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Chapter 5**

**Abstract**

Visualization

*and Osvaldo A. Rosso*

of the nonlinear systems.

distribution function

**1. Introduction**

**85**

Informational Time Causal Planes:

A Tool for Chaotic Map Dynamic

In the present chapter, we made a detailed analysis of the different regimes of certain chaotic systems and their correspondence with the change in the normalized

information measure can be applied to follow the changes in the behavior variations

**Keywords:** chaotic dynamics, statistical complexity, information theory quantifiers,

In the space of few decades, chaos theory has jumped from the scientific literature into the popular realm, being regarded as a new way of looking at complex systems like brains or ecosystems. It is believed that the theory manages to capture the disorganized order that pervades our world. Chaos theory is a facet of the complex system paradigm having to do with determinism randomness. As many other people before, we wish to approach it from the information theory viewpoint. In 1959 Kolmogorov had pointed out that the probabilistic theory of information developed by Shannon could be applied to symbolic encodings of the phase space descriptions of physical non-linear dynamical systems and with line of rezoning it more or less direct characterize a process in terms of *its Kolmogorov-Sinai entropy* [1, 2]. It has been a cornerstone in the updated theory of dynamical systems that could be complimented with Pesin's theorem in 1977 [3]. With this theorem, Pesin has proven that for certain deterministic nonlinear dynamical systems exhibiting chaotic behavior, an estimation of the *Kolmogorov-Sinai entropy* can be computed as

As is well known, chaotic systems have sensitivity to initial conditions which means instability everywhere in the phase space and lead to nonperiodic motion

Shannon entropy, Statistical Complexity, and Fisher information measure. We construct a bidimensional plane composed of the selection of a pair of the informational tools mentioned above (a casual plane is defined), in which the different dynamical regimes appeared very clear and give more information of the underlying process. In such a way, a plane composed of the normalized Shannon

entropy, statistical complexity, normalized Shannon entropy, and Fisher

Shannon entropy, Fisher information measure, Bandt-Pompe probability

the sum of the positive Lyapunov exponents for the process.

*Felipe Olivares, Lindiane Souza, Walter Legnani*

[38] Biazar J, Asadi MA, Salehi F. Rational Homotopy perturbation method for solving stiff systems of ordinary differential equations. Applied Mathematical Modelling. 2015;**39**: 12911299

#### **Chapter 5**

[38] Biazar J, Asadi MA, Salehi F. Rational Homotopy perturbation method for solving stiff systems of ordinary differential equations. Applied Mathematical Modelling. 2015;**39**:

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

12911299

**84**

## Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization

*Felipe Olivares, Lindiane Souza, Walter Legnani and Osvaldo A. Rosso*

#### **Abstract**

In the present chapter, we made a detailed analysis of the different regimes of certain chaotic systems and their correspondence with the change in the normalized Shannon entropy, Statistical Complexity, and Fisher information measure. We construct a bidimensional plane composed of the selection of a pair of the informational tools mentioned above (a casual plane is defined), in which the different dynamical regimes appeared very clear and give more information of the underlying process. In such a way, a plane composed of the normalized Shannon entropy, statistical complexity, normalized Shannon entropy, and Fisher information measure can be applied to follow the changes in the behavior variations of the nonlinear systems.

**Keywords:** chaotic dynamics, statistical complexity, information theory quantifiers, Shannon entropy, Fisher information measure, Bandt-Pompe probability distribution function

#### **1. Introduction**

In the space of few decades, chaos theory has jumped from the scientific literature into the popular realm, being regarded as a new way of looking at complex systems like brains or ecosystems. It is believed that the theory manages to capture the disorganized order that pervades our world. Chaos theory is a facet of the complex system paradigm having to do with determinism randomness. As many other people before, we wish to approach it from the information theory viewpoint.

In 1959 Kolmogorov had pointed out that the probabilistic theory of information developed by Shannon could be applied to symbolic encodings of the phase space descriptions of physical non-linear dynamical systems and with line of rezoning it more or less direct characterize a process in terms of *its Kolmogorov-Sinai entropy* [1, 2]. It has been a cornerstone in the updated theory of dynamical systems that could be complimented with Pesin's theorem in 1977 [3]. With this theorem, Pesin has proven that for certain deterministic nonlinear dynamical systems exhibiting chaotic behavior, an estimation of the *Kolmogorov-Sinai entropy* can be computed as the sum of the positive Lyapunov exponents for the process.

As is well known, chaotic systems have sensitivity to initial conditions which means instability everywhere in the phase space and lead to nonperiodic motion

(chaotic time series) [4]. One of the main characteristics of this kind of systems is its capability of long-term unpredictability despite the deterministic character of the temporal trajectory. In a system undergoing chaotic motion, two closeup neighboring points in the phase space after a short time elapsed show an exponential divergence of their respective trajectories. For example, let *X*1ð Þ*t* and *X*2ð Þ*t* be such two points, located within a ball of radius *R* at time *t*. Further, assume that these two points cannot be resolved within the ball due to poor instrumental resolution. At some later time *t* 0 , the distance between the points will typically grow to *X*<sup>1</sup> *t* <sup>0</sup> ð Þ� *X*<sup>2</sup> *t* <sup>0</sup> j j ð Þ ≈j j *X*1ðÞ�*t X*2ð Þ*t* exp Λ *t* ð Þ j j <sup>0</sup> � *t* , in the case of chaotic dynamics, with Λ . 0, the average of Lyapunov exponents of the system. Clearly, if *X*<sup>1</sup> *t* <sup>0</sup> ð Þ� *X*<sup>2</sup> *t* <sup>0</sup> j j ð Þ . *R*, the points will be apart from each other, determining a nonzero distance between them. This fact could be interpreted by a certain kind of instability which reveals some information about the phase space population that was not available at earlier times [4]. This fact contributes to think that the chaotic behavior plays a role of *information source*.

interpreted as a measure of uncertainty. The *Shannon entropy* can be considered as

This concept means a global measure of the information contained in the TS; it has a low degree of sensitivity to strong changes in the distribution originating from

For a time series XðÞ� *t xt* f g ; *t* ¼ 1*;* …*; M* , a set of *M* measures of the observable

*N*

*i*¼1 *pi* ln *pi*

equal to zero when the outcomes of a certain experiment denoted by the index *k* associated with probabilities *pk* ≈1 will occur. Therefore, the known dynamics developed by the dynamical system under study is complete. If the knowledge of the system dynamics is minimal, all the states of the system can occur with equal probability; thus, this probability can be modeled by a uniform distribution

It is useful to define the so-called normalized Shannon entropy, denoted as *H*[*P*]

In order to analyze the local aspects of variations in the content of information given by a TS is extended the use of the Fisher's Information Measure, which uses the gradient content of the PDF, and a difference that means the Shannon Entropy, the FIM as can be seen from its definition given in the expression (4) reflect tiny

*=ρ*ð Þ *x dx* ¼ 4

In this sense, the Fisher information is a local information quantifier. It has various interpretations, and, among others, it can be thought of as a measure of the ability to estimate a parameter. In other cases, it is applied to calculate the amount of information that can be extracted from a TS and also as a measure of the state of disorder of a system or phenomenon [8]. The so-called Cramer-Rao bound can be considered as the most important property in which the FIM participates [9]. The local sensitivity of FIM can contribute in such cases in which the analysis necessitates an appeal to a notion of *order*. When there are certain points in the set Ω at which the PDF *ρ*ð Þ! *x* 0 is convenient to redefine the FIM

ð Ω

∂ <sup>∂</sup>*<sup>x</sup>* ½ � *<sup>ψ</sup>*ð Þ *<sup>x</sup>*

� � � �

2

*dx,* (4)

� � � �

the number of possible states of the system under study, the *Shannon entropy*

*S P*½ �¼�<sup>X</sup>

<sup>Ω</sup> *ρ*ð Þ *x dx* ¼ 1; its

*<sup>i</sup>*¼<sup>1</sup> *pi* <sup>¼</sup> 1 and *<sup>N</sup>* as

*ρ*ð Þ *x* ln ð Þ *ρ*ð Þ *x dx:* (1)

� �*:* (2)

; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �*,* which is

; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, with <sup>P</sup>*<sup>N</sup>*

*H P*½ �¼ *S P*½ �*=Smax,* (3)

one of the most representative examples of information quantifiers. Let a continuous PDF be noted by *<sup>ρ</sup>*ð Þ *<sup>x</sup>* with *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup> and <sup>Ð</sup>

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

*S*½ �¼� *ρ*

(formally *Shannon's logarithmic information*) [7] is defined by

Eq. (2) constitutes a function of the probability *P* ¼ *pi*

ð Ω

associated *Shannon Entropy S*½ � *ρ* is defined by [7]:

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

a small-sized region of the set Ω.

*Pe* <sup>¼</sup> *pi* <sup>¼</sup> <sup>1</sup>*=N*; <sup>∀</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �.

(0 ≤ *H*[*P*] ≤ 1) with *Smax* ¼ *S Pe* ½ �¼ ln *N*.

ð Ω

∂ <sup>∂</sup>*<sup>x</sup>* ½ � *<sup>ρ</sup>*ð Þ *<sup>x</sup>*

� � � �

2

� � � �

localized perturbations. It reads [8, 9]

*F*½ �¼ *ρ*

*<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>p</sup> .

where *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* ffiffiffiffiffiffiffiffiffi

**87**

in which its expression is

X and the associated PDF, given by *P* ¼ *pi*

As has been shown in the literature for a many of simple nonlinear processes, the Lyapunov exponents may be computed very precisely with different algorithms. In such a way, a nonlinear dynamical system may be considered as an information source from which information-related quantifiers may help visualize relevant details of the chaotic process. The existence of simple "calibrated" sources such as the logistic map provides a means for a precise evaluation of the performance of these information quantifiers. In this communication we take advantage such fact in order to show that planar representations constructed with two information theory-based quantifiers offer one possibility of easily visualizing many interesting details of chaos characteristics, including the fine structure of chaotic attractors. We exemplified their use showing the result on two chaotic maps: the logistic map and the delayed logistic map.

#### **2. Information theory quantifier prescription**

Many systems during its functioning generate a sequence of values that can be measured constituting what is called in science as time series (TS). The analysis concerns to extract the major quantity of information of them to accomplish the understanding of the meaning of the changes characterizing different dynamical regimes. It usually computes the experimental, or when the case permits the theoretical, probability distribution function (PDF) of the regimes exhibited by the TS, from here noted as Xð Þ*t* .

The mathematical tools applied once the PDF is available receive the name of informational tools; more precisely information theory quantifiers [5], the main feature of the quantifiers is exactly quantifying the amount of information coming from the TS, originating in the dynamical system.

#### **2.1 Shannon entropy, Fisher information measure, and statistical complexity**

The concept of entropy has many interpretations arising from a wide diversity of scientific and technological fields. Among them is associated with disorder, with the volume of state space, and with a lack of information too. There are various definitions according to ways of computing this important magnitude to study the dynamics of the systems, and one of the most frequent that could be considered of foundational definition is the denominated *Shannon entropy* [6], which can be

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

interpreted as a measure of uncertainty. The *Shannon entropy* can be considered as one of the most representative examples of information quantifiers.

Let a continuous PDF be noted by *<sup>ρ</sup>*ð Þ *<sup>x</sup>* with *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> <sup>⊂</sup> <sup>R</sup> and <sup>Ð</sup> <sup>Ω</sup> *ρ*ð Þ *x dx* ¼ 1; its associated *Shannon Entropy S*½ � *ρ* is defined by [7]:

$$\mathcal{S}[\rho] = -\int\_{\Omega} \rho(\mathbf{x}) \ln \left( \rho(\mathbf{x}) \right) d\mathbf{x}.\tag{1}$$

This concept means a global measure of the information contained in the TS; it has a low degree of sensitivity to strong changes in the distribution originating from a small-sized region of the set Ω.

For a time series XðÞ� *t xt* f g ; *t* ¼ 1*;* …*; M* , a set of *M* measures of the observable X and the associated PDF, given by *P* ¼ *pi* ; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, with <sup>P</sup>*<sup>N</sup> <sup>i</sup>*¼<sup>1</sup> *pi* <sup>¼</sup> 1 and *<sup>N</sup>* as the number of possible states of the system under study, the *Shannon entropy* (formally *Shannon's logarithmic information*) [7] is defined by

$$S[P] = -\sum\_{i=1}^{N} p\_i \ln \left( p\_i \right). \tag{2}$$

Eq. (2) constitutes a function of the probability *P* ¼ *pi* ; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �*,* which is equal to zero when the outcomes of a certain experiment denoted by the index *k* associated with probabilities *pk* ≈1 will occur. Therefore, the known dynamics developed by the dynamical system under study is complete. If the knowledge of the system dynamics is minimal, all the states of the system can occur with equal probability; thus, this probability can be modeled by a uniform distribution *Pe* <sup>¼</sup> *pi* <sup>¼</sup> <sup>1</sup>*=N*; <sup>∀</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �.

It is useful to define the so-called normalized Shannon entropy, denoted as *H*[*P*] in which its expression is

$$H[P] = \mathbb{S}[P] / \mathbb{S}\_{\text{max}},\tag{3}$$

(0 ≤ *H*[*P*] ≤ 1) with *Smax* ¼ *S Pe* ½ �¼ ln *N*.

In order to analyze the local aspects of variations in the content of information given by a TS is extended the use of the Fisher's Information Measure, which uses the gradient content of the PDF, and a difference that means the Shannon Entropy, the FIM as can be seen from its definition given in the expression (4) reflect tiny localized perturbations. It reads [8, 9]

$$F[\rho] = \int\_{\Omega} \left| \frac{\partial}{\partial \mathbf{x}} [\rho(\mathbf{x})] \right|^2 / \rho(\mathbf{x}) d\mathbf{x} = 4 \int\_{\Omega} \left| \frac{\partial}{\partial \mathbf{x}} [\rho(\mathbf{x})] \right|^2 d\mathbf{x},\tag{4}$$

where *<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* ffiffiffiffiffiffiffiffiffi *<sup>ρ</sup>*ð Þ *<sup>x</sup>* <sup>p</sup> .

In this sense, the Fisher information is a local information quantifier. It has various interpretations, and, among others, it can be thought of as a measure of the ability to estimate a parameter. In other cases, it is applied to calculate the amount of information that can be extracted from a TS and also as a measure of the state of disorder of a system or phenomenon [8]. The so-called Cramer-Rao bound can be considered as the most important property in which the FIM participates [9]. The local sensitivity of FIM can contribute in such cases in which the analysis necessitates an appeal to a notion of *order*. When there are certain points in the set Ω at which the PDF *ρ*ð Þ! *x* 0 is convenient to redefine the FIM

(chaotic time series) [4]. One of the main characteristics of this kind of systems is its capability of long-term unpredictability despite the deterministic character of the temporal trajectory. In a system undergoing chaotic motion, two closeup neighboring points in the phase space after a short time elapsed show an exponential divergence of their respective trajectories. For example, let *X*1ð Þ*t* and *X*2ð Þ*t* be such two points, located within a ball of radius *R* at time *t*. Further, assume that these two points cannot be resolved within the ball due to poor instrumental resolution. At

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

, the distance between the points will typically grow to

<sup>0</sup> j j ð Þ ≈j j *X*1ðÞ�*t X*2ð Þ*t* exp Λ *t* ð Þ j j <sup>0</sup> � *t* , in the case of chaotic dynamics, with Λ . 0, the average of Lyapunov exponents of the system. Clearly, if

<sup>0</sup> j j ð Þ . *R*, the points will be apart from each other, determining a nonzero distance between them. This fact could be interpreted by a certain kind of instability which reveals some information about the phase space population that was not available at earlier times [4]. This fact contributes to think that the chaotic

As has been shown in the literature for a many of simple nonlinear processes, the Lyapunov exponents may be computed very precisely with different algorithms. In such a way, a nonlinear dynamical system may be considered as an information source from which information-related quantifiers may help visualize relevant details of the chaotic process. The existence of simple "calibrated" sources such as the logistic map provides a means for a precise evaluation of the performance of these information quantifiers. In this communication we take advantage such fact in order to show that planar representations constructed with two information theory-based quantifiers offer one possibility of easily visualizing many interesting details of chaos characteristics, including the fine structure of chaotic attractors. We exemplified their use showing the result on two chaotic maps: the logistic map

Many systems during its functioning generate a sequence of values that can be measured constituting what is called in science as time series (TS). The analysis concerns to extract the major quantity of information of them to accomplish the understanding of the meaning of the changes characterizing different dynamical regimes. It usually computes the experimental, or when the case permits the theoretical, probability distribution function (PDF) of the regimes exhibited by the TS,

The mathematical tools applied once the PDF is available receive the name of informational tools; more precisely information theory quantifiers [5], the main feature of the quantifiers is exactly quantifying the amount of information coming

**2.1 Shannon entropy, Fisher information measure, and statistical complexity**

The concept of entropy has many interpretations arising from a wide diversity of scientific and technological fields. Among them is associated with disorder, with the volume of state space, and with a lack of information too. There are various definitions according to ways of computing this important magnitude to study the dynamics of the systems, and one of the most frequent that could be considered of foundational definition is the denominated *Shannon entropy* [6], which can be

some later time *t*

<sup>0</sup> ð Þ� *X*<sup>2</sup> *t*

<sup>0</sup> ð Þ� *X*<sup>2</sup> *t*

*X*<sup>1</sup> *t*

*X*<sup>1</sup> *t*

0

and the delayed logistic map.

from here noted as Xð Þ*t* .

**86**

**2. Information theory quantifier prescription**

from the TS, originating in the dynamical system.

behavior plays a role of *information source*.

avoiding the division by *ρ*ð Þ *x* , in such cases an alternative expression of can be found in [9].

The signal discretization carries a problem of loss of information. It was extended studies by several authors, for example, see [10, 11] and references therein. In particular, it entails the loss of Fisher's shift invariance, which has not been relevant in the present chapter. Taking in mind the considerations made above, the discrete normalized FIM runs over the interval [0,1] and [12] is given by

$$F[P] = F\_0 \sum\_{i=1}^{N-1} \left[ \left( p\_{i+1} \right)^{1/2} - \left( p\_i \right)^{1/2} \right]^2,\tag{5}$$

In this way, we have 0 ≤ *H P*½ �≤ 1 and 0≤ *QJ P; Pe* ½ �≤1.

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

**2.2 The Bandt and Pompe approach to building up a PDF**

algorithm given by the statistics in the literature to do with this task.

quantified by a no-null *C*[*P*].

functional form adopted by *H* and *QJ*.

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

in the usual amplitude statistic methodology.

physical problem.

ered of the form

**89**

The *C P*½ � quantifies the existence of correlational structures giving a measure of the complexity of a TS. In the case of perfect order or total randomness of a signal coming of a dynamical system, the value of the *C*[*P*] is identically null that means the signal possesses no structure. In between these two extreme instances, a large range of possible stages of physical structure may be realized by a dynamical system. These stages should be reflected in the features of the obtained PDF and

The global character of the SCM arising in that its value does not change with different orderings of the PDF. So the *C P*½ � quantifies the disorder but also the degree of correlational structures. It is evident that the SCM adopted in this work is a not a trivial function of the entropy. It has consequences in the ranges that this information quantifier can take. For a given *H* value, the complexity *C* runs on a precise range limited by a minimum *Cmin* and a maximum *Cmax* [16]. These extreme values depend only on the probability space dimension and, of course, on the

In the beginning of this section, it was mentioned that during the analysis of a TS, one of the first steps is the computation of the PDF associated. Immediately a question emerges: What is the appropriate PDF that can be computed from the TS? The regrettable answer is not unique. There is no universal nonparametric

To give light in this subject, Bandt and Pompe (BP) [17] introduce a simple and robust symbolic method that takes into account the time causality connected with dynamics of the system. They proposed to use a symbol sequence from the TS that can be constructed in a natural way. So the PDF introduced by Bandt and Pompe (BP-PDF) did not use any kind of assumption about the model, in general

unknown, in which of the underlying dynamics exists. To compute the BP-PDF, the "partitions" are constructed by comparing the order of neighboring relative values in the TS rather than by apportioning amplitudes according to different levels like

One problem remains linked with the lack of information associated with the temporal causality in which origins are in the computed methodologies to calculate the amplitude of the histograms. To give an answer to this problem, Kowalski and co-workers [18] using the Cressie-Read family of divergence measure showed in quantitative assessment the advantages of the BP-PDF in relation to any scheme based upon the construction of the corresponding amplitude histogram of the PDF, and also the BP-PDF brought some insight information about the dynamics of the

Two parameters are necessary to define at the time of computing the BP-PDF, namely, the embedding dimension and the embedding delay. To clarify these crucial concepts, we will give the following details. Let TS XðÞ¼ *t xt* f g ; *t* ¼ 1*;* …*; M* , with an embedding dimension *D* . 1 (*D* ∈ N) and an embedding delay *τ* . 1 (*τ* ∈ N); the BP pattern of order *D* generated by this selection of parameters shall be consid-

*s*↦ *xs*�ð Þ *<sup>D</sup>*�<sup>1</sup> *<sup>τ</sup>; xs*�ð Þ *<sup>D</sup>*�<sup>2</sup> *<sup>τ</sup>; xs*�ð Þ *<sup>D</sup>*�<sup>3</sup> *<sup>τ</sup>;* …*; xs*�*<sup>τ</sup>; xs*

So the methodology proposed by Bandt and Pompe has as a starting point for every time *s,* assigned with a *D*-dimensional vector that results from the evaluation

*:* (10)

where the normalization constant *F*<sup>0</sup> is given by

$$F\_0 = \begin{cases} \mathbf{1} & \text{if } p\_{i^\*} = \mathbf{1} for \, i^\* = \mathbf{1} or \, i^\* = \mathbf{N} \, and \, p\_i = \mathbf{0} \forall i \neq i^\* \\ \mathbf{1}/2 & \text{otherwise} \end{cases} \tag{6}$$

The local sensitivity of FIM for discrete PDFs is reflected by the fact that the specific *i-*ordering of the discrete values in *P* ¼ *pi* ; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � � must be seriously taken into account in evaluating the sum in Eq. (5) [13]. Each term in Eq. (5) can be regarded as a kind of "distance" between two contiguous probabilities. Thus, a different ordering of the pertinent summands would lead to a different FIM value, thereby its local nature.

In a system with *N* different states which reach a very ordered state, we can think it generates a signal with a PDF given by *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *pk* ffi <sup>1</sup>*;* and *pi* ffi <sup>0</sup>; <sup>∀</sup>*<sup>k</sup>* 6¼ � *i* ¼ 1*;* …*; N*g, as it has a Shannon entropy *S P*½ �ffi <sup>0</sup> 0 and a normalized FIM *F P*½ �ffi <sup>0</sup> *F*<sup>0</sup> ¼ 1*:* In the other extreme, if the system under analysis develops a very disordered state, it is natural to assume that this particular state is described by a PDF approximated by a uniform distribution *Pe* <sup>¼</sup> *pi* <sup>¼</sup> <sup>1</sup>*=N*; <sup>∀</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, and the corresponding Shannon entropy *S Pe* ½ �ffi *Smax* ¼ ln *N* while *F P*½ �ffi <sup>0</sup> 0. In certain way it is easy to understand that the general behavior of the FIM is opposite to that of the Shannon entropy.

The third information quantifier applied in this chapter is the *statistical complexity measure* (SCM) which is a global informational quantifier. All the computations made in the present work were done with the definitions introduced by López-Ruiz et al., in their seminal paper [14] with improvements advanced by Lamberti et al. [15]. For a discrete probability distribution function (PDF), *P* ¼ *pi* ; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, associated with a time series (TS), this functional *C P*½ � is given by

$$C[P] = Q\_{\!\!\!/ }[P, P\_{\!\!\!/ }].H[P]. \tag{7}$$

where *H* denotes the amount of "disorder" given by the normalized Shannon entropy (Eq. (3)) and *QJ* is called "disequilibrium," defined in terms of the Jensen-Shannon divergence, given by

$$Q\_{\!\!\!/}[P, P\_{\epsilon}] = Q\_0 J[P, P\_{\epsilon}] = Q\_0 \{ \mathbf{S}[(P + P\_{\epsilon})/2] - \mathbf{S}[P]/2 - \mathbf{S}[P\_{\epsilon}]/2 \}. \tag{8}$$

The normalization condition *Q*<sup>0</sup> for the disequilibrium corresponds to the inverse of the maximum possible value of Jensen-Shannon divergence, that is, *Q*<sup>0</sup> ¼ *J P*0*; Pe* ½ �:

$$Q\_0 = -2\left\{ \left(\frac{N+1}{N}\right) \ln\left(N+1\right) - \ln\left(2N\right) + \ln N \right\}^{-1}.\tag{9}$$

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

In this way, we have 0 ≤ *H P*½ �≤ 1 and 0≤ *QJ P; Pe* ½ �≤1.

avoiding the division by *ρ*ð Þ *x* , in such cases an alternative expression of can be

The signal discretization carries a problem of loss of information. It was extended studies by several authors, for example, see [10, 11] and references therein. In particular, it entails the loss of Fisher's shift invariance, which has not been relevant in the present chapter. Taking in mind the considerations made above, the discrete normalized FIM runs over the interval [0,1] and [12] is given by

> *pi*þ<sup>1</sup> � �1*=*<sup>2</sup> � *pi* � �1*=*<sup>2</sup> h i<sup>2</sup>

<sup>1</sup>*=*<sup>2</sup> *otherwise* **.**

The local sensitivity of FIM for discrete PDFs is reflected by the fact that the

taken into account in evaluating the sum in Eq. (5) [13]. Each term in Eq. (5) can be regarded as a kind of "distance" between two contiguous probabilities. Thus, a different ordering of the pertinent summands would lead to a different FIM value,

In a system with *N* different states which reach a very ordered state, we can think it generates a signal with a PDF given by *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> *pk* ffi <sup>1</sup>*;* and *pi* ffi <sup>0</sup>; <sup>∀</sup>*<sup>k</sup>* 6¼ � *i* ¼ 1*;* …*; N*g, as it has a Shannon entropy *S P*½ �ffi <sup>0</sup> 0 and a normalized FIM

*F P*½ �ffi <sup>0</sup> *F*<sup>0</sup> ¼ 1*:* In the other extreme, if the system under analysis develops a very disordered state, it is natural to assume that this particular state is described by a PDF approximated by a uniform distribution *Pe* <sup>¼</sup> *pi* <sup>¼</sup> <sup>1</sup>*=N*; <sup>∀</sup>*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, and the corresponding Shannon entropy *S Pe* ½ �ffi *Smax* ¼ ln *N* while *F P*½ �ffi <sup>0</sup> 0. In certain way it is easy to understand that the general behavior of the FIM is opposite to that

The third information quantifier applied in this chapter is the *statistical complexity measure* (SCM) which is a global informational quantifier. All the computations made in the present work were done with the definitions introduced by López-Ruiz et al., in their seminal paper [14] with improvements advanced by Lamberti et al. [15]. For

where *H* denotes the amount of "disorder" given by the normalized Shannon entropy (Eq. (3)) and *QJ* is called "disequilibrium," defined in terms of the Jensen-

The normalization condition *Q*<sup>0</sup> for the disequilibrium corresponds to the inverse of the maximum possible value of Jensen-Shannon divergence, that is,

*QJ P; Pe* ½ �¼ *Q*<sup>0</sup> *J P; Pe* ½ �¼ *Q*<sup>0</sup> *S P*½ �� ð Þ þ *Pe =*2 *S P*½ �*=*2 � *S Pe* f g ½ �*=*2 *:* (8)

ln ð Þ� *N* þ 1 ln 2ð Þþ *N* ln *N* � ��<sup>1</sup>

*,* (5)

; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � � must be seriously

; *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*;* …*; <sup>N</sup>* � �, associated

*:* (9)

*C P*½ �¼ *QJ P; Pe* ½ �*:H P*½ �*,* (7)

(6)

*F P*½ �¼ *F*<sup>0</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

where the normalization constant *F*<sup>0</sup> is given by

specific *i-*ordering of the discrete values in *P* ¼ *pi*

a discrete probability distribution function (PDF), *P* ¼ *pi*

*N* þ 1 *N* � �

with a time series (TS), this functional *C P*½ � is given by

X *N*�1

*i*¼1

*<sup>F</sup>*<sup>0</sup> <sup>¼</sup> <sup>1</sup> *if pi* <sup>∗</sup> <sup>¼</sup> <sup>1</sup> *for i* <sup>∗</sup> <sup>¼</sup> <sup>1</sup> *or i* <sup>∗</sup> <sup>¼</sup> *N and pi* <sup>¼</sup> <sup>0</sup>∀*<sup>i</sup>* 6¼ *<sup>i</sup>* <sup>∗</sup>

found in [9].

�

thereby its local nature.

of the Shannon entropy.

Shannon divergence, given by

*Q*<sup>0</sup> ¼ �2

*Q*<sup>0</sup> ¼ *J P*0*; Pe* ½ �:

**88**

The *C P*½ � quantifies the existence of correlational structures giving a measure of the complexity of a TS. In the case of perfect order or total randomness of a signal coming of a dynamical system, the value of the *C*[*P*] is identically null that means the signal possesses no structure. In between these two extreme instances, a large range of possible stages of physical structure may be realized by a dynamical system. These stages should be reflected in the features of the obtained PDF and quantified by a no-null *C*[*P*].

The global character of the SCM arising in that its value does not change with different orderings of the PDF. So the *C P*½ � quantifies the disorder but also the degree of correlational structures. It is evident that the SCM adopted in this work is a not a trivial function of the entropy. It has consequences in the ranges that this information quantifier can take. For a given *H* value, the complexity *C* runs on a precise range limited by a minimum *Cmin* and a maximum *Cmax* [16]. These extreme values depend only on the probability space dimension and, of course, on the functional form adopted by *H* and *QJ*.

#### **2.2 The Bandt and Pompe approach to building up a PDF**

In the beginning of this section, it was mentioned that during the analysis of a TS, one of the first steps is the computation of the PDF associated. Immediately a question emerges: What is the appropriate PDF that can be computed from the TS? The regrettable answer is not unique. There is no universal nonparametric algorithm given by the statistics in the literature to do with this task.

To give light in this subject, Bandt and Pompe (BP) [17] introduce a simple and robust symbolic method that takes into account the time causality connected with dynamics of the system. They proposed to use a symbol sequence from the TS that can be constructed in a natural way. So the PDF introduced by Bandt and Pompe (BP-PDF) did not use any kind of assumption about the model, in general unknown, in which of the underlying dynamics exists. To compute the BP-PDF, the "partitions" are constructed by comparing the order of neighboring relative values in the TS rather than by apportioning amplitudes according to different levels like in the usual amplitude statistic methodology.

One problem remains linked with the lack of information associated with the temporal causality in which origins are in the computed methodologies to calculate the amplitude of the histograms. To give an answer to this problem, Kowalski and co-workers [18] using the Cressie-Read family of divergence measure showed in quantitative assessment the advantages of the BP-PDF in relation to any scheme based upon the construction of the corresponding amplitude histogram of the PDF, and also the BP-PDF brought some insight information about the dynamics of the physical problem.

Two parameters are necessary to define at the time of computing the BP-PDF, namely, the embedding dimension and the embedding delay. To clarify these crucial concepts, we will give the following details. Let TS XðÞ¼ *t xt* f g ; *t* ¼ 1*;* …*; M* , with an embedding dimension *D* . 1 (*D* ∈ N) and an embedding delay *τ* . 1 (*τ* ∈ N); the BP pattern of order *D* generated by this selection of parameters shall be considered of the form

$$\mathfrak{s} \mapsto \left( \mathfrak{x}\_{\mathfrak{s}-(D-1)\mathfrak{r}}, \mathfrak{x}\_{\mathfrak{s}-(D-2)\mathfrak{r}}, \mathfrak{x}\_{\mathfrak{s}-(D-3)\mathfrak{r}}, \dots, \mathfrak{x}\_{\mathfrak{s}-\mathfrak{r}}, \mathfrak{x}\_{\mathfrak{s}} \right). \tag{10}$$

So the methodology proposed by Bandt and Pompe has as a starting point for every time *s,* assigned with a *D*-dimensional vector that results from the evaluation of Xð Þ*t* at times *s* � ð Þ *D* � 1 *τ, s* � ð Þ *D* � 2 *τ,* …*, s* � *τ, s*. It is easy to note that higher values of *D* imply more information about "the past" to contribute in the PDF.

Once time settled the ordinal pattern of order *D* related to the time sequence *s,* the next step is to compute the permutation pattern denoted by *π* ¼ ð Þ *r*0*;r*1*;* …*;rD*�<sup>1</sup> of 0ð Þ *;* 1*;* …*; D* � 1 that could be formalized by

$$\mathbb{1}\mathbb{X}\_{\mathfrak{t}-r\_{(D-1)}\mathfrak{r}} \le \mathbb{x}\_{\mathfrak{t}-r\_{(D-2)}\mathfrak{r}} \le \dots \le \mathbb{x}\_{\mathfrak{t}-r\_{1}\mathfrak{r}} \le \mathbb{x}\_{\mathfrak{t}-r\_{0}\mathfrak{r}}.\tag{11}$$

**2.3 Ordinal patterns for deterministic processes**

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

called causality Shannon-Fisher plane (*H* � *F*), respectively.

**2.4 Causal informational planes**

**3. Description of the chaotic maps**

logistic map with delay.

order difference equation:

numerically [26] via

**91**

**3.1 The logistic map**

There is a demonstrated fact, done by Amigó et al. [21, 22], that in the case of deterministic one-dimensional maps, independently of the TS length M, *not all possible ordinal patterns*, applying BP methodology [17], can effectively give orbits in the phase space. This is a kind of a new dynamical property that means the existence of *forbidden ordinal patterns.* The proximity of patterns as well as correlation is not linked with the abovementioned property [21, 22]. So the informational quantifiers give a new characteristic in the analysis of chaotic or deterministic TS.

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

To characterize a given dynamical system described by a TS, we are able to use two representation spaces: (a) one with global-global characteristics called causal entropy-complexity plane (*H* � *C*Þ and (b) one with global-local characteristics

The time causal nature of the Bandt and Pompe PDF gives a criterion to separate and differentiate chaotic and stochastic systems in different regions in both informational planes (*H*½ �� Π *C*½ �Þ Π [23] and (*H*½ �� Π *F*½ �Þ Π [24, 25]. While the global plane gives information of the complexity of a system, the local one becomes able to

separate different dynamical behaviors in function of a control parameter.

We focus our attention on two chaotic maps, namely, the logistic map and

One of the most used examples of deterministic chaotic systems is the logistic map. Its simplicity and easy computational implementation had been one of the most useful tools to explain chaotic behavior. It is a quadratic map F : *xn* ! *xn*þ<sup>1</sup> [26], described by the ecologically motivated, dissipative system given by the first-

where 0 ≤*xn* ≤ 1 and 0 ≤*r*≤4 can be associated with a kind of growth rate in the

population dynamics. The corresponding Lyapunov exponent can be evaluated

1 *N* X *N*�1

*n*¼0

where *N* is the number of iterations. **Figure 1a** and **1b** displays the well-known bifurcation diagram and the corresponding Lyapunov exponent Λð Þ*r* , respectively, as a function of the parameter 3*:*4≤ *r*≤4*:*0 with *Δr* ¼ 0*:*0005*:* We evaluated numerically the logistic map starting from a random initial condition in the interval <sup>0</sup> , *<sup>x</sup>*<sup>0</sup> , <sup>0</sup>*:*5. The first *<sup>N</sup>*<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup> iterations are disregarded (transitory states), and the next *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> ones are used for Lyapunov evaluation (Eq. (14)) and information

Λð Þ¼ *r* lim *N*!∞

theory quantifiers (Eqs. (3), (5), and (7)).

*xn*þ<sup>1</sup> ¼ *r xn*ð Þ 1 � *xn ,* (13)

ln j j *r*ð Þ 1 � 2*xn ,* (14)

At this stage of the BP-PDF procedure, the vector defined by Eq. (10) is converted into a definite symbol *π*. Then to get a unique result, BP considers that *rk* , *rk*�<sup>1</sup> if *xs*�*rk<sup>τ</sup>* ¼ *xs*�*rk*�1*<sup>τ</sup>*. This is justified if the values of f g *xt* have a continuous distribution so that equal values are very unusual.

Considering all the *D*! possible orderings (permutations) *π<sup>i</sup>* when embedding dimension *D*, their associated relative frequencies can be naturally computed according to the number of times; this sequence order is found in the TS, divided by the total number of sequences

$$p(\pi\_i) = \frac{\#\{s \mid \mathbf{s} \le M - (D - 1)\tau; (s) \text{ has type } \pi\_i\}}{M - (D - 1)\tau}. \tag{12}$$

In Eq. (12) the symbol # (usually applied to designate the set cardinality) means "number." In such a way, an ordinal pattern probability distribution Π ¼ *p*ð Þ *π<sup>i</sup>* f g ; *i* ¼ 1*;* …*; D*! is constructed from the TS.

Time series amplitude information is not considered, and it is a clear disadvantage of the methodology proposed by BP, but it is compensated by the valuable information given by the intrinsic structure of the process under analysis. The scheme proposed by BP can be understood as a symbolic representation of time series by recourse to a comparison of consecutive points (*τ* ¼ 1) or nonconsecutive (*τ* . 1) points allowing for an accurate empirical reconstruction of the underlying phase space, even in the presence of weak (observational and dynamical) noise [17]. It is noticeable that the ordinal-pattern's associated PDF results invariant with respect to nonlinear monotonous transformations. Accordingly, nonlinear drifts or scaling artificially introduced by a measurement device will not modify the quantifier estimation, a nice property if one deals with experimental data (see [19]). Summing up all these advantages makes the BP methodology a better choice than conventional methods based on range partitioning.

Among other properties, we can mention the following characteristics to give reasons in the selection of the BP-PDF: (i) the reduced number of parameters needed contributes to its simplicity of implementation (*D* and *τ* the embedding length and delay, respectively), and (ii) the time required in the calculation process is in fact very short. The BP methodology has an extra advantage; it can be used to compute the PDF in TS arising in low-dimensional dynamical systems, and signals originated in a wide diversity of systems as well as chaotic, noisy, and regular reality-based ones, with a light analysis in the stationarity because there no mandatory condition to accomplish with a strong stationary assumption (for details see [17]).

Parameter *D*, required by the BP-PDF methodology, determines the number of accessible states which is given by *D!*. Moreover, the minimum length of the TS must satisfy the condition *M* >> *D!* in order to achieve a reliable statistic and proper distinction between stochastic and deterministic dynamics [20]. The seminal work of BP [17] includes an advice on the choice of range of the parameters to compute the BP-PDF, when the selection of time lag is *τ* ¼ 1, and recommends the other parameter (*D*) to pick up on the interval 3≤ *D* ≤ 6*:*

#### **2.3 Ordinal patterns for deterministic processes**

There is a demonstrated fact, done by Amigó et al. [21, 22], that in the case of deterministic one-dimensional maps, independently of the TS length M, *not all possible ordinal patterns*, applying BP methodology [17], can effectively give orbits in the phase space. This is a kind of a new dynamical property that means the existence of *forbidden ordinal patterns.* The proximity of patterns as well as correlation is not linked with the abovementioned property [21, 22]. So the informational quantifiers give a new characteristic in the analysis of chaotic or deterministic TS.

#### **2.4 Causal informational planes**

of Xð Þ*t* at times *s* � ð Þ *D* � 1 *τ, s* � ð Þ *D* � 2 *τ,* …*, s* � *τ, s*. It is easy to note that higher values of *D* imply more information about "the past" to contribute in the PDF. Once time settled the ordinal pattern of order *D* related to the time sequence *s,* the next step is to compute the permutation pattern denoted by *π* ¼ ð Þ *r*0*;r*1*;* …*;rD*�<sup>1</sup>

At this stage of the BP-PDF procedure, the vector defined by Eq. (10) is converted into a definite symbol *π*. Then to get a unique result, BP considers that *rk* , *rk*�<sup>1</sup> if *xs*�*rk<sup>τ</sup>* ¼ *xs*�*rk*�1*<sup>τ</sup>*. This is justified if the values of f g *xt* have a continuous

Considering all the *D*! possible orderings (permutations) *π<sup>i</sup>* when embedding dimension *D*, their associated relative frequencies can be naturally computed according to the number of times; this sequence order is found in the TS, divided by

#f g *s*j*s*≤ *M* � ð Þ *D* � 1 *τ; s*ð Þ *has type π<sup>i</sup>*

In Eq. (12) the symbol # (usually applied to designate the set cardinality) means

Time series amplitude information is not considered, and it is a clear disadvantage of the methodology proposed by BP, but it is compensated by the valuable information given by the intrinsic structure of the process under analysis. The scheme proposed by BP can be understood as a symbolic representation of time series by recourse to a comparison of consecutive points (*τ* ¼ 1) or nonconsecutive (*τ* . 1) points allowing for an accurate empirical reconstruction of the underlying phase space, even in the presence of weak (observational and dynamical) noise [17]. It is noticeable that the ordinal-pattern's associated PDF results invariant with respect to nonlinear monotonous transformations. Accordingly, nonlinear drifts or scaling artificially introduced by a measurement device will not modify the quantifier estimation, a nice property if one deals with experimental data (see [19]). Summing up all these advantages makes the BP methodology a better choice than

Among other properties, we can mention the following characteristics to give reasons in the selection of the BP-PDF: (i) the reduced number of parameters needed contributes to its simplicity of implementation (*D* and *τ* the embedding length and delay, respectively), and (ii) the time required in the calculation process is in fact very short. The BP methodology has an extra advantage; it can be used to compute the PDF in TS arising in low-dimensional dynamical systems, and signals originated in a wide diversity of systems as well as chaotic, noisy, and regular reality-based ones, with a light analysis in the stationarity because there no

mandatory condition to accomplish with a strong stationary assumption (for details

Parameter *D*, required by the BP-PDF methodology, determines the number of accessible states which is given by *D!*. Moreover, the minimum length of the TS must satisfy the condition *M* >> *D!* in order to achieve a reliable statistic and proper distinction between stochastic and deterministic dynamics [20]. The seminal work of BP [17] includes an advice on the choice of range of the parameters to compute the BP-PDF, when the selection of time lag is *τ* ¼ 1, and recommends the other

"number." In such a way, an ordinal pattern probability distribution

*xs*�*r*ð Þ *<sup>D</sup>*�<sup>1</sup> *<sup>τ</sup>* ≤ *xs*�*r*ð Þ *<sup>D</sup>*�<sup>2</sup> *<sup>τ</sup>* ≤ … ≤*xs*�*r*1*<sup>τ</sup>* ≤*xs*�*r*0*<sup>τ</sup>:* (11)

*<sup>M</sup>* � ð Þ *<sup>D</sup>* � <sup>1</sup> *<sup>τ</sup> :* (12)

of 0ð Þ *;* 1*;* …*; D* � 1 that could be formalized by

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

distribution so that equal values are very unusual.

*p*ð Þ¼ *π<sup>i</sup>*

Π ¼ *p*ð Þ *π<sup>i</sup>* f g ; *i* ¼ 1*;* …*; D*! is constructed from the TS.

conventional methods based on range partitioning.

parameter (*D*) to pick up on the interval 3≤ *D* ≤ 6*:*

see [17]).

**90**

the total number of sequences

To characterize a given dynamical system described by a TS, we are able to use two representation spaces: (a) one with global-global characteristics called causal entropy-complexity plane (*H* � *C*Þ and (b) one with global-local characteristics called causality Shannon-Fisher plane (*H* � *F*), respectively.

The time causal nature of the Bandt and Pompe PDF gives a criterion to separate and differentiate chaotic and stochastic systems in different regions in both informational planes (*H*½ �� Π *C*½ �Þ Π [23] and (*H*½ �� Π *F*½ �Þ Π [24, 25]. While the global plane gives information of the complexity of a system, the local one becomes able to separate different dynamical behaviors in function of a control parameter.

#### **3. Description of the chaotic maps**

We focus our attention on two chaotic maps, namely, the logistic map and logistic map with delay.

#### **3.1 The logistic map**

One of the most used examples of deterministic chaotic systems is the logistic map. Its simplicity and easy computational implementation had been one of the most useful tools to explain chaotic behavior. It is a quadratic map F : *xn* ! *xn*þ<sup>1</sup> [26], described by the ecologically motivated, dissipative system given by the firstorder difference equation:

$$\boldsymbol{\infty}\_{n+1} = r \,\,\boldsymbol{\infty}\_{n} (1 - \boldsymbol{\pi}\_{n}),\tag{13}$$

where 0 ≤*xn* ≤ 1 and 0 ≤*r*≤4 can be associated with a kind of growth rate in the population dynamics. The corresponding Lyapunov exponent can be evaluated numerically [26] via

$$\Lambda(r) = \lim\_{N \to \infty} \frac{1}{N} \sum\_{n=0}^{N-1} \ln|r(1 - 2\mathbf{x}\_n)|,\tag{14}$$

where *N* is the number of iterations. **Figure 1a** and **1b** displays the well-known bifurcation diagram and the corresponding Lyapunov exponent Λð Þ*r* , respectively, as a function of the parameter 3*:*4≤ *r*≤4*:*0 with *Δr* ¼ 0*:*0005*:* We evaluated numerically the logistic map starting from a random initial condition in the interval <sup>0</sup> , *<sup>x</sup>*<sup>0</sup> , <sup>0</sup>*:*5. The first *<sup>N</sup>*<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup> iterations are disregarded (transitory states), and the next *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> ones are used for Lyapunov evaluation (Eq. (14)) and information theory quantifiers (Eqs. (3), (5), and (7)).

infinitely long, and finite regions of the interval are visited by the orbits. Many periodic windows are observed, and all possible periods are represented, but the width of the window decreases as the period increases. Periodic windows suddenly appear as *r* increases, and they contain their own periodic-doubling route toward

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

In **Figure 1a** and **1b**, we marked eight zones in order to analyze the logistic map behavior. They are *Zone 1*, *r*∈ ½ Þ 3*:*4*;r*<sup>∞</sup> which corresponds to the period-doubling zone; *Zone 2*, *r*∈½ Þ *r*∞*;r*<sup>1</sup> , with *r*<sup>1</sup> ¼ 3*:*626557*,* which corresponds to the start of periodic window of period 6; *Zone 3*, *r*∈½ Þ *r*1*;r*<sup>2</sup> , with *r*<sup>2</sup> ¼ 3*:*701645*,* which corresponds to the start of periodic window of period 7; *Zone 4*, *r*∈ ½ Þ *r*2*;r*<sup>3</sup> , with

*r*<sup>3</sup> ¼ 3*:*738177*,* which corresponds to the start of periodic window of period 5; *Zone 5*, *r*∈½ Þ *r*3*;r*<sup>4</sup> , with *r*<sup>4</sup> ¼ 3*:*828427*,* which corresponds to the start of periodic window

*r*<sup>5</sup> ¼ 3*:*905573*,* which corresponds to the start of periodic window of period 5; *Zone 7*, *r*∈½ Þ *r*5*;r*<sup>6</sup> , with *r*<sup>6</sup> ¼ 3*:*960108*,* which corresponds to the start of periodic window of period 4; and *Zone 8*, *r*∈½ � *r*6*;* 4 , with *r* ¼ 4 for fully developed chaos. Periodic windows "interrupt" chaotic behavior in noticeable fashion. At the beginning of a window, there is a sudden and dramatic change in the long-term behavior of the logistic map. Consider, for example, the behavior for *r*≥*r*<sup>4</sup> corresponding to the beginning of a period 3 window. We see three miniature copies of the whole final-state diagram (**Figure 1a**), and, indeed, we can reproduce the entire scenario of *period-doubling* ! *chaos* (*band splitting*) ! *chaos* (*band merging*) again, albeit at a much smaller scale. Same findings are encountered at all the other periodic windows, including miniature windows within the larger windows,

In 1948 Hutchinson [27] introduces a delay in the logistic equation to improve its applications in the study of population dynamics. The proposed model by Hutchinson has been applied in population dynamics [28], deterministic chaotic systems [29], the analysis of random discrete delay equations [30–32], etc. We face a dis-

with 0≤ *Xn* ≤1 and 0≤ *r*≤2*:*3 (*r* the intrinsic growth). The equation resembles the logistic map (Eq. (13)) saved for the fact that the factor regulating population

It is convenient to convert the second-order difference equation into an equivalent pair of first-order difference equations. The logistic map with delay is thus

*xn*þ<sup>1</sup> ¼ *r xn* 1 � *yn*

and the corresponding Lyapunov exponents can be evaluated numerically

� � � � �

*yn*þ<sup>1</sup> <sup>¼</sup> *xn*

(

� �

ln *<sup>r</sup>*<sup>2</sup> <sup>1</sup> � *yn* � *xn zn*

*,*

� �<sup>2</sup> <sup>þ</sup> <sup>1</sup> 1 þ *z*<sup>2</sup> *n*

� � � � �

*,* (17)

(16)

*Xn*þ<sup>1</sup> ¼ *r Xn*ð Þ 1 � *Xn*�<sup>1</sup> *,* (15)

crete logistic equation with delay [33] given by the difference equation:

chaos. These facts exhibit the self-similar nature of the logistic map.

of period 3; *Zone 6*, *r*∈ ½ Þ *r*4*;r*<sup>5</sup> , the largest periodic window, with

as evidence of self-similarity.

**3.2 The logistic map with delay**

growth contains a one-generation time delay.

Λ1ð Þ¼ *r* lim

*N*!∞

1 2*N* X *N*�1

*n*¼0

recasted as a two-dimensional map:

[26] via

**93**

#### **Figure 1.**

*(a) Bifurcation diagram and (b) Lyapunov exponent Λ for the logistic map as function of parameter r (Δr* ¼ *0:0005). The vertical segment lines delimited the different dynamical windows described in the text.*

In the bifurcation diagram (**Figure 1a**), for fixed *r*, one appreciates that a periodic orbit consists of a countable set of points, while a chaotic attractor fills out dense bands within the unit interval. For *r*∈½ Þ 0*;* 1 , one detects stable behavior *xn* ¼ 0. For *r*∈½ Þ 1*;* 3 , there exist only a single steady-state solution given by *xn* ¼ 1 � 1*=r*. Increasing the control parameter, for *r*∈½ Þ 3*;r*<sup>∞</sup> , forces the system to undergo period-doubling bifurcations. Cycles of periods 2*,* 4*,* 8*,* 16*,* 32*,* etc. occur, and, if *rn* denotes the values of *r* for which a 2*<sup>n</sup>* cycle first appears, succesive *rn*s converge to the limiting value *r*<sup>∞</sup> ≈ 3*:*5699456 [26]. The value *r*<sup>∞</sup> splits the finalstate diagram into two distinct parts: (a) the period-doubling zone on the left and (b) an area governed mainly by increasing chaotic behavior on the right. From **Figure 1b**, we see that period-doubling zone *r*∈½ Þ 3*;r*<sup>∞</sup> Lyapunov are Λð Þ*r* ≤0*,* approximating to zero at each period-doubling bifurcation. The onset of chaos is apparent at *r*<sup>∞</sup> where Λ becomes positive for the first time. For *r* ¼ 4 the iterates of the logistic map are represented by a random-looking distribution of dots which vertically span the range *xn* ∈½ � 0*;* 1 , that is, complete developed chaos. For *r* . *r*<sup>∞</sup> the Lyapunov exponent increases globally (see **Figure 1b**), except for dips one sees in the windows of periodic behavior. In the chaotic regime *r*∈½ � *r*∞*;* 4 *,* the period is

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

infinitely long, and finite regions of the interval are visited by the orbits. Many periodic windows are observed, and all possible periods are represented, but the width of the window decreases as the period increases. Periodic windows suddenly appear as *r* increases, and they contain their own periodic-doubling route toward chaos. These facts exhibit the self-similar nature of the logistic map.

In **Figure 1a** and **1b**, we marked eight zones in order to analyze the logistic map behavior. They are *Zone 1*, *r*∈ ½ Þ 3*:*4*;r*<sup>∞</sup> which corresponds to the period-doubling zone; *Zone 2*, *r*∈½ Þ *r*∞*;r*<sup>1</sup> , with *r*<sup>1</sup> ¼ 3*:*626557*,* which corresponds to the start of periodic window of period 6; *Zone 3*, *r*∈½ Þ *r*1*;r*<sup>2</sup> , with *r*<sup>2</sup> ¼ 3*:*701645*,* which corresponds to the start of periodic window of period 7; *Zone 4*, *r*∈ ½ Þ *r*2*;r*<sup>3</sup> , with *r*<sup>3</sup> ¼ 3*:*738177*,* which corresponds to the start of periodic window of period 5; *Zone 5*, *r*∈½ Þ *r*3*;r*<sup>4</sup> , with *r*<sup>4</sup> ¼ 3*:*828427*,* which corresponds to the start of periodic window of period 3; *Zone 6*, *r*∈ ½ Þ *r*4*;r*<sup>5</sup> , the largest periodic window, with *r*<sup>5</sup> ¼ 3*:*905573*,* which corresponds to the start of periodic window of period 5; *Zone 7*, *r*∈½ Þ *r*5*;r*<sup>6</sup> , with *r*<sup>6</sup> ¼ 3*:*960108*,* which corresponds to the start of periodic window of period 4; and *Zone 8*, *r*∈½ � *r*6*;* 4 , with *r* ¼ 4 for fully developed chaos.

Periodic windows "interrupt" chaotic behavior in noticeable fashion. At the beginning of a window, there is a sudden and dramatic change in the long-term behavior of the logistic map. Consider, for example, the behavior for *r*≥*r*<sup>4</sup> corresponding to the beginning of a period 3 window. We see three miniature copies of the whole final-state diagram (**Figure 1a**), and, indeed, we can reproduce the entire scenario of *period-doubling* ! *chaos* (*band splitting*) ! *chaos* (*band merging*) again, albeit at a much smaller scale. Same findings are encountered at all the other periodic windows, including miniature windows within the larger windows, as evidence of self-similarity.

#### **3.2 The logistic map with delay**

In 1948 Hutchinson [27] introduces a delay in the logistic equation to improve its applications in the study of population dynamics. The proposed model by Hutchinson has been applied in population dynamics [28], deterministic chaotic systems [29], the analysis of random discrete delay equations [30–32], etc. We face a discrete logistic equation with delay [33] given by the difference equation:

$$X\_{n+1} = r \, X\_n (1 - X\_{n-1}),\tag{15}$$

with 0≤ *Xn* ≤1 and 0≤ *r*≤2*:*3 (*r* the intrinsic growth). The equation resembles the logistic map (Eq. (13)) saved for the fact that the factor regulating population growth contains a one-generation time delay.

It is convenient to convert the second-order difference equation into an equivalent pair of first-order difference equations. The logistic map with delay is thus recasted as a two-dimensional map:

$$\begin{cases} \varkappa\_{n+1} = r \varkappa\_n \left( 1 - \wp\_n \right) \\\\ \varkappa\_{n+1} = \varkappa\_n \end{cases},\tag{16}$$

and the corresponding Lyapunov exponents can be evaluated numerically [26] via

$$\Lambda\_1(r) = \lim\_{N \to \infty} \frac{1}{2N} \sum\_{n=0}^{N-1} \ln \left| \frac{r^2 \left(\mathbf{1} - \mathbf{y}\_n - \mathbf{x}\_n \, \mathbf{z}\_n\right)^2 + \mathbf{1}}{\mathbf{1} + \mathbf{z}\_n^2} \right|, \tag{17}$$

In the bifurcation diagram (**Figure 1a**), for fixed *r*, one appreciates that a periodic orbit consists of a countable set of points, while a chaotic attractor fills out dense bands within the unit interval. For *r*∈½ Þ 0*;* 1 , one detects stable behavior *xn* ¼ 0. For *r*∈½ Þ 1*;* 3 , there exist only a single steady-state solution given by *xn* ¼ 1 � 1*=r*. Increasing the control parameter, for *r*∈½ Þ 3*;r*<sup>∞</sup> , forces the system to undergo period-doubling bifurcations. Cycles of periods 2*,* 4*,* 8*,* 16*,* 32*,* etc. occur, and, if *rn* denotes the values of *r* for which a 2*<sup>n</sup>* cycle first appears, succesive *rn*s converge to the limiting value *r*<sup>∞</sup> ≈ 3*:*5699456 [26]. The value *r*<sup>∞</sup> splits the finalstate diagram into two distinct parts: (a) the period-doubling zone on the left and (b) an area governed mainly by increasing chaotic behavior on the right. From **Figure 1b**, we see that period-doubling zone *r*∈½ Þ 3*;r*<sup>∞</sup> Lyapunov are Λð Þ*r* ≤0*,* approximating to zero at each period-doubling bifurcation. The onset of chaos is apparent at *r*<sup>∞</sup> where Λ becomes positive for the first time. For *r* ¼ 4 the iterates of the logistic map are represented by a random-looking distribution of dots which vertically span the range *xn* ∈½ � 0*;* 1 , that is, complete developed chaos. For *r* . *r*<sup>∞</sup> the Lyapunov exponent increases globally (see **Figure 1b**), except for dips one sees in the windows of periodic behavior. In the chaotic regime *r*∈½ � *r*∞*;* 4 *,* the period is

*(a) Bifurcation diagram and (b) Lyapunov exponent Λ for the logistic map as function of parameter r (Δr* ¼ *0:0005). The vertical segment lines delimited the different dynamical windows described in the text.*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Figure 1.**

**92**

and

$$
\Lambda\_1(r) + \Lambda\_2(r) = \lim\_{N \to \infty} \frac{1}{N} \sum\_{n=0}^{N-1} \ln|r \propto\_n|\tag{18}
$$

This map has common characteristics with the usual logistic map. In particular,

Lyapunov exponents (see **Figure 2b**) indicates quasiperiodicity in the region where

For each chaotic map previously described, the same time series of length *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> data, used for evaluating the corresponding Lyapunov exponents at each parameter value, are used now to build a Bandt-Pompe PDF (Π), taking an embedding dimension *D* ¼ 6 and time lag *τ* ¼ 1. Then corresponding time causal information theory quantifiers, normalized Shannon entropy (*H*½ �Þ Π , statistical complexity (*C*½ � Π ), and Fisher information measure (*F*½ � Π ), were evaluated.

For ordinal entropic quantifiers of Shannon kind (global quantifiers), the BP-PDF provides univocal prescription. However, some ambiguities arise in the case in which one wishes to employ the BP-PDF to construct local quantifiers. The local sensitivity of the Fisher information measure for discrete PDFs is reflected in the fact that the specific "*i*-ordering" of the discrete values *p*ð Þ *π<sup>i</sup>* must be taken into account in evaluating Eq. (5). If we are working with BP-PDF and consider patterns of length *D*, we will have *D*!! possibilities for the *i*-ordering. We follow the Lehmer lexicographic order [34] in the generation of BP-PDF, because it provides the best graphic separation of different dynamics in the causal Shannon-Fisher plane [13, 24]. We display in **Figures 3** and **4** the causal information quantifiers (entropy, complexity and Fisher) as a function of the parameter *r* for the logistic map (see **Figure 1**) and delayed logistic map (see **Figure 2**), respectively. In these figures, the different dynamical zones and corresponding colors are both used in the original

For the logistic map, the period-doubling zone is detected for all the quantifiers. In particular for *r* , *r*∞, low entropy and complexity values and maximum Fisher value are found with the different periodic behaviors. A jump in the entropy and complexity value and a drop in Fisher value are observed when period doubling happens. This quantifier behavior is due to for periodic sequences the BP-PDF

After *r*<sup>∞</sup> the dynamic becomes chaotic (positive Lyapunov exponent). An abrupt entropy and complexity growth and Fisher decreasing values are observed for *r* . *r*<sup>∞</sup> reaching their maximum value at *r* ¼ 4*,* where we face a totally developed chaotic dynamics. The several "drops" in the entropy and complexity, with the "jumps" in the Fisher values in the parameter interval *r*<sup>∞</sup> , *r*≤ 4, correspond to the periodic windows as can be easily confirmed compared with the bifurcation and

For the delayed logistic map, a regular dynamic (steady state) is observed for parameter in the range *r* , *rH* and then has entropy and complexity null values and Fisher maximum value. For *r* . *rH* an oscillatory behavior appears, which is Hopf

Λ<sup>1</sup> ¼ 0.

**4. Results and discussion**

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

bifurcation diagrams (**Figures 1a** and **2a**).

consisting of a very few *p*ð Þ *π<sup>i</sup>* 6¼ 0 values.

Lyapunov exponent (see **Figure 1**).

**95**

*X* ¼ 0 is a fixed point for *r*∈½ Þ 0*;* 1 , and *Xn* ¼ 1 � 1*=r* is a stationary state for ∈ ½ Þ 1*;rH* . In the delayed logistic map case, when the parameter value is *rH* ¼ 2, the system shows a Poincare-Andronov-Hopf bifurcation (see **Figure 2**). The quasiperiodic behavior persists over most of the range *r*∈ ½ Þ *rH;r*<sup>1</sup> . A seven-cycle periodicity is observed for *r*∈ ½ Þ *r*1*;r*<sup>2</sup> (with *r*<sup>1</sup> ¼ 2*:*17640 and *r*<sup>2</sup> ¼ 2*:*20071). For the parameter *r*∈ *r*2*;rc* ½ Þ, one mainly detects chaotic dynamics interspersed with regions of relative simplicity. For *r* . *rc* ¼ 2*:*271, the finite solutions are destabilized, and the system experiences a transition to �∞*:* In the bifurcation diagram (see **Figure 2a**), it is difficult to distinguish quasiperiodicity from chaos, but the plot displaying

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

with *zn*þ<sup>1</sup> ¼ 1*= r* 1 � *yn* � *xnzn* � � �. In the previous equations, *N* is the number of iterations.

The pertinent bifurcation diagram and the corresponding Lyapunov exponents Λ<sup>1</sup> and Λ<sup>2</sup> are displayed in **Figure 2a** and **2b**, as a function of the parameter 0≤*r*≤2*:*3 with *Δr* ¼ 0*:*0005*,*respectively. We evaluated numerically the delayed logistic map starting from a random initial condition. The first *<sup>N</sup>*<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup> iterations are disregarded (transitory states), and the next *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> ones are used for Lyapunov evaluation (Eqs. (15) and (16)) and information theory quantifiers (Eqs. (3), (5) and (7)).

**Figure 2.**

*(a) Bifurcation diagram and (b) Lyapunov exponents Λ<sup>1</sup> and Λ<sup>2</sup> for the delayed logistic map as function of parameter r (Δr* ¼ *0:0005). The vertical segment lines delimited the different dynamical windows described in the text.*

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

This map has common characteristics with the usual logistic map. In particular, *X* ¼ 0 is a fixed point for *r*∈½ Þ 0*;* 1 , and *Xn* ¼ 1 � 1*=r* is a stationary state for ∈ ½ Þ 1*;rH* . In the delayed logistic map case, when the parameter value is *rH* ¼ 2, the system shows a Poincare-Andronov-Hopf bifurcation (see **Figure 2**). The quasiperiodic behavior persists over most of the range *r*∈ ½ Þ *rH;r*<sup>1</sup> . A seven-cycle periodicity is observed for *r*∈ ½ Þ *r*1*;r*<sup>2</sup> (with *r*<sup>1</sup> ¼ 2*:*17640 and *r*<sup>2</sup> ¼ 2*:*20071). For the parameter *r*∈ *r*2*;rc* ½ Þ, one mainly detects chaotic dynamics interspersed with regions of relative simplicity. For *r* . *rc* ¼ 2*:*271, the finite solutions are destabilized, and the system experiences a transition to �∞*:* In the bifurcation diagram (see **Figure 2a**), it is difficult to distinguish quasiperiodicity from chaos, but the plot displaying Lyapunov exponents (see **Figure 2b**) indicates quasiperiodicity in the region where Λ<sup>1</sup> ¼ 0.

#### **4. Results and discussion**

and

iterations.

**Figure 2.**

*the text.*

**94**

(Eqs. (3), (5) and (7)).

with *zn*þ<sup>1</sup> ¼ 1*= r* 1 � *yn* � *xnzn*

Λ1ð Þþ *r* Λ2ð Þ¼ *r* lim

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*N*!∞

The pertinent bifurcation diagram and the corresponding Lyapunov exponents

Λ<sup>1</sup> and Λ<sup>2</sup> are displayed in **Figure 2a** and **2b**, as a function of the parameter 0≤*r*≤2*:*3 with *Δr* ¼ 0*:*0005*,*respectively. We evaluated numerically the delayed logistic map starting from a random initial condition. The first *<sup>N</sup>*<sup>0</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup> iterations

are disregarded (transitory states), and the next *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> ones are used for Lyapunov evaluation (Eqs. (15) and (16)) and information theory quantifiers

*(a) Bifurcation diagram and (b) Lyapunov exponents Λ<sup>1</sup> and Λ<sup>2</sup> for the delayed logistic map as function of parameter r (Δr* ¼ *0:0005). The vertical segment lines delimited the different dynamical windows described in*

1 *N* X *N*�1

*n*¼0

� � �. In the previous equations, *N* is the number of

ln j j *r xn ,* (18)

For each chaotic map previously described, the same time series of length *<sup>N</sup>* <sup>¼</sup> <sup>10</sup><sup>6</sup> data, used for evaluating the corresponding Lyapunov exponents at each parameter value, are used now to build a Bandt-Pompe PDF (Π), taking an embedding dimension *D* ¼ 6 and time lag *τ* ¼ 1. Then corresponding time causal information theory quantifiers, normalized Shannon entropy (*H*½ �Þ Π , statistical complexity (*C*½ � Π ), and Fisher information measure (*F*½ � Π ), were evaluated.

For ordinal entropic quantifiers of Shannon kind (global quantifiers), the BP-PDF provides univocal prescription. However, some ambiguities arise in the case in which one wishes to employ the BP-PDF to construct local quantifiers. The local sensitivity of the Fisher information measure for discrete PDFs is reflected in the fact that the specific "*i*-ordering" of the discrete values *p*ð Þ *π<sup>i</sup>* must be taken into account in evaluating Eq. (5). If we are working with BP-PDF and consider patterns of length *D*, we will have *D*!! possibilities for the *i*-ordering. We follow the Lehmer lexicographic order [34] in the generation of BP-PDF, because it provides the best graphic separation of different dynamics in the causal Shannon-Fisher plane [13, 24]. We display in **Figures 3** and **4** the causal information quantifiers (entropy, complexity and Fisher) as a function of the parameter *r* for the logistic map (see **Figure 1**) and delayed logistic map (see **Figure 2**), respectively. In these figures, the different dynamical zones and corresponding colors are both used in the original bifurcation diagrams (**Figures 1a** and **2a**).

For the logistic map, the period-doubling zone is detected for all the quantifiers. In particular for *r* , *r*∞, low entropy and complexity values and maximum Fisher value are found with the different periodic behaviors. A jump in the entropy and complexity value and a drop in Fisher value are observed when period doubling happens. This quantifier behavior is due to for periodic sequences the BP-PDF consisting of a very few *p*ð Þ *π<sup>i</sup>* 6¼ 0 values.

After *r*<sup>∞</sup> the dynamic becomes chaotic (positive Lyapunov exponent). An abrupt entropy and complexity growth and Fisher decreasing values are observed for *r* . *r*<sup>∞</sup> reaching their maximum value at *r* ¼ 4*,* where we face a totally developed chaotic dynamics. The several "drops" in the entropy and complexity, with the "jumps" in the Fisher values in the parameter interval *r*<sup>∞</sup> , *r*≤ 4, correspond to the periodic windows as can be easily confirmed compared with the bifurcation and Lyapunov exponent (see **Figure 1**).

For the delayed logistic map, a regular dynamic (steady state) is observed for parameter in the range *r* , *rH* and then has entropy and complexity null values and Fisher maximum value. For *r* . *rH* an oscillatory behavior appears, which is Hopf

#### **Figure 3.**

*Time causal information quantifiers for logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *as a function of the parameter r (Δr* ¼ *0:0005): (a) italicized Shannon entropy; (b) statistical complexity; and (c) Fisher information measure, evaluated with Bandt-Pompe PDF, with D* ¼ *6, τ* ¼ *1:The vertical segment lines delimited the different dynamical windows described in the text. The color code for the different zones is the same as in Figure 1a.*

**Figure 4.**

**97**

*same as in Figure 2a.*

*Time causal information quantifiers for delayed logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *as function of the parameter r (Δr* ¼ *0:0005): (a) italicized Shannon entropy; (b) statistical complexity; and (c) Fisher information measure, evaluated with Bandt-Pompe PDF, with D* ¼ *6 and τ* ¼ *1: The vertical segment lines delimited the different dynamical windows described in the text. The color code for the different zones is the*

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

#### **Figure 4.**

**Figure 3.**

*Figure 1a.*

**96**

*Time causal information quantifiers for logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *as a function of the parameter r (Δr* ¼ *0:0005): (a) italicized Shannon entropy; (b) statistical complexity; and (c) Fisher information measure, evaluated with Bandt-Pompe PDF, with D* ¼ *6, τ* ¼ *1:The vertical segment lines delimited the different dynamical windows described in the text. The color code for the different zones is the same as in*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Time causal information quantifiers for delayed logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *as function of the parameter r (Δr* ¼ *0:0005): (a) italicized Shannon entropy; (b) statistical complexity; and (c) Fisher information measure, evaluated with Bandt-Pompe PDF, with D* ¼ *6 and τ* ¼ *1: The vertical segment lines delimited the different dynamical windows described in the text. The color code for the different zones is the same as in Figure 2a.*

bifurcation with Λ<sup>1</sup> ¼ 0. This quasiperiodic orbit can be thought as a mixture of periodic orbits of several different fundamental frequencies. The three quantifiers, entropy, complexity, and Fisher, are able to detect changes in the quasiperiodicity oscillations as a function of *r,* being Fisher the most sensitive (see **Figure 3b**). The growth in the amplitude of this oscillatory behavior as a function *r* is not detected by the quantifiers because of the independence of the BP-PDF on the amplitude values. Note that the same number of ordinal patterns (�30 patterns) is materialized for the whole quasiperiodic behavior, indicating its deterministic nature but not giving

indications about the type of dynamics. For the parameter values *r*<sup>1</sup> ≤ *r*≤ *r*2*,* a period 7 window with *H* ¼ 0*:*299576*, C* ¼ 0*:*288624, and *F* ¼ 1 is observed. In the parameter range *r*<sup>2</sup> ≤*r* , *rc*, chaotic dynamics with some periodic windows is observed, characterized by higher values of entropy and complexity and lower values of Fisher,

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

The causal planes *H* � *C* and *H* � *F* for the logistic map in the parameter range 3*:*4≤*r*≤4*:*0 are shown in **Figure 5a** and **5b**, respectively. Both planes provide a

*Time causal information planes for delayed logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *for parameter <sup>r</sup> (Δr* ¼ *0:0005): (a) causal entropy-complexity plane and (b) causal Shannon-Fisher plane. The color code for*

in relation to those previously obtained for the period 7 window.

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

**Figure 6.**

**99**

*the different zones is the same as in Figure 2a.*

#### **Figure 5.**

*Time causal information planes for logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *for parameter <sup>r</sup> (Δ<sup>r</sup>* <sup>¼</sup> *<sup>0</sup>:0005): (a) causal entropy-complexity plane and (b) causal Shannon-Fisher plane. The color code for the different zones is the same as in Figure 1a.*

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

indications about the type of dynamics. For the parameter values *r*<sup>1</sup> ≤ *r*≤ *r*2*,* a period 7 window with *H* ¼ 0*:*299576*, C* ¼ 0*:*288624, and *F* ¼ 1 is observed. In the parameter range *r*<sup>2</sup> ≤*r* , *rc*, chaotic dynamics with some periodic windows is observed, characterized by higher values of entropy and complexity and lower values of Fisher, in relation to those previously obtained for the period 7 window.

The causal planes *H* � *C* and *H* � *F* for the logistic map in the parameter range 3*:*4≤*r*≤4*:*0 are shown in **Figure 5a** and **5b**, respectively. Both planes provide a

#### **Figure 6.**

*Time causal information planes for delayed logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *for parameter <sup>r</sup> (Δr* ¼ *0:0005): (a) causal entropy-complexity plane and (b) causal Shannon-Fisher plane. The color code for the different zones is the same as in Figure 2a.*

bifurcation with Λ<sup>1</sup> ¼ 0. This quasiperiodic orbit can be thought as a mixture of periodic orbits of several different fundamental frequencies. The three quantifiers, entropy, complexity, and Fisher, are able to detect changes in the quasiperiodicity oscillations as a function of *r,* being Fisher the most sensitive (see **Figure 3b**). The growth in the amplitude of this oscillatory behavior as a function *r* is not detected by the quantifiers because of the independence of the BP-PDF on the amplitude values. Note that the same number of ordinal patterns (�30 patterns) is materialized for the whole quasiperiodic behavior, indicating its deterministic nature but not giving

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*Time causal information planes for logistic map time series (<sup>M</sup>* <sup>¼</sup> *106 data*<sup>Þ</sup> *for parameter <sup>r</sup> (Δ<sup>r</sup>* <sup>¼</sup> *<sup>0</sup>:0005): (a) causal entropy-complexity plane and (b) causal Shannon-Fisher plane. The color code for the different*

**Figure 5.**

**98**

*zones is the same as in Figure 1a.*

characterization of the intrinsic information of the system, independently of the control parameter. From the causal plane *H* � *C* (**Figure 5a**), we can observed that the variation in the whole range of the parameter *r* locates the system very close to the maximum complexity curve *Cmax*, reaching at *r* ¼ 4 (totally developed chaos) and its maximum value *C* ¼ 0*:*48425. Note also that low entropy values *H* , 0*:*3 correspond to periodic behavior values; however, these values have also associated high values of complexity with curve *Cmax*, making difficult in this way the clear separation of the different dynamic behaviors, but a quantification of the global complexity of the logistic map is obtained. The causal plane *H* � *F* (see **Figure 5b**) shows a clear characterization of the various associated dynamics to different values of the control parameter *r*, locating them in different zones of the plane. It is in this instance, in which the Fisher permutation reveals its local character and, simultaneous with the global information delivered by Shannon entropy, gives us a sort of "topographical" plane of the dynamics.

In **Figure 6**, we show the behavior of the delayed logistic map for the whole range of the parameter *r* in the two causality planes. It is clearly that the *H* � *C* plane (**Figure 6a**) gives us just the information of the complexity of the map, which reaches the maximum curve (*Cmax*), but does not differentiate between a Hopf bifurcation and the chaotic dynamics developed for *r* . *r*2. On the other hand in the *H* � *F* plane (**Figure 6b**), one obtains a good defined structure for the quasiperiodic orbits, due to the oscillations in the quantifiers. The shapeless blue points for *r*<sup>2</sup> , *r* , *rc* is due to neither the *H* and the *F* are not detecting any kind of intermittency or bifurcation that can be present into the chaotic dynamic, at the present parameter resolution Δ*r:*

#### **5. Conclusions**

We have shown that taken as starting point a probabilistic description of dynamical system considering the inherent temporal causality in the generated time series throughout Bandt-Pompe methodology, it is possible to evaluate information quantifiers of global or local character and a complete and detailed characterization of the dynamical system can be successfully archived with reference to an information causal plane, in which the two coordinate axes are different information quantifiers. The causal information planes defined are the global-global *H* � *C* plane and the global-local *H* � *F* plane, in which (i) the permutation normalized Shannon entropy ð Þ *H*½ � Π and the permutation statistical complexity ð Þ *C*½ � Π are responsible for the global features and (ii) the permutation Fisher information measure ð Þ *F*½ � Π accounts for the local attributes (all the information quantifiers are evaluated using BP-PDF denoted by Π).

**Author details**

Felipe Olivares<sup>1</sup>

Valparaíso, Chile

Argentina

Argentina

**101**

, Lindiane Souza<sup>2</sup>

\*Address all correspondence to: oarosso@gmail.com

provided the original work is properly cited.

1 Instituto Física, Pontificia Universidad Católica de Valparaíso (PUCV),

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization*

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

2 Instituto Física, Universidad Federal de Alagoas (UFAL), Maceió, Brazil

3 Signals and Images Processsing Center (CSPSI), Facultad Regional Buenos Aires, Universidad Tecnológica Naciona (UTN), Ciudad Autónoma de Buenos Aires,

4 Instituto de Medicina Traslacional e Ingeniería Biomedica (IMTIB), Hospital Italiano de Buenos Aires (HIBA), CONICET, Ciudad Autónoma de Buenos Aires,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

, Walter Legnani<sup>3</sup> and Osvaldo A. Rosso2,4\*

For the discrete systems considered here, the logistic map and the delayed logistic map, we find that both *H* and *C* show a correspondence with one of the classic measures of chaoticity, the maximum exponent of Lyapunov, while the local sensitivity of *F* reveals details of the dynamics, invisible to the other quantifiers. The visualization of the location of the dynamics of the system under analysis, in the information planes, allows us to account for (a) the complexity of the system and (b) characterization of different dynamics in different locations of the plane, enabling the identification of different routes to chaos.

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

#### **Author details**

characterization of the intrinsic information of the system, independently of the control parameter. From the causal plane *H* � *C* (**Figure 5a**), we can observed that the variation in the whole range of the parameter *r* locates the system very close to the maximum complexity curve *Cmax*, reaching at *r* ¼ 4 (totally developed chaos)

*H* , 0*:*3 correspond to periodic behavior values; however, these values have also associated high values of complexity with curve *Cmax*, making difficult in this way the clear separation of the different dynamic behaviors, but a quantification of the global complexity of the logistic map is obtained. The causal plane *H* � *F* (see **Figure 5b**) shows a clear characterization of the various associated dynamics to different values of the control parameter *r*, locating them in different zones of the plane. It is in this instance, in which the Fisher permutation reveals its local character and, simultaneous with the global information delivered by Shannon entropy,

In **Figure 6**, we show the behavior of the delayed logistic map for the whole range of the parameter *r* in the two causality planes. It is clearly that the *H* � *C* plane (**Figure 6a**) gives us just the information of the complexity of the map, which reaches the maximum curve (*Cmax*), but does not differentiate between a Hopf bifurcation and the chaotic dynamics developed for *r* . *r*2. On the other hand in the *H* � *F* plane (**Figure 6b**), one obtains a good defined structure for the quasiperiodic orbits, due to the oscillations in the quantifiers. The shapeless blue points for *r*<sup>2</sup> , *r* , *rc* is due to neither the *H* and the *F* are not detecting any kind of intermittency or bifurcation that can be present into the chaotic dynamic, at the present

We have shown that taken as starting point a probabilistic description of dynamical system considering the inherent temporal causality in the generated time series throughout Bandt-Pompe methodology, it is possible to evaluate information quantifiers of global or local character and a complete and detailed characterization of the dynamical system can be successfully archived with reference to an information causal plane, in which the two coordinate axes are different information quantifiers. The causal information planes defined are the global-global *H* � *C* plane and the global-local *H* � *F* plane, in which (i) the permutation normalized Shannon entropy ð Þ *H*½ � Π and the permutation statistical complexity ð Þ *C*½ � Π are responsible for the global features and (ii) the permutation Fisher information measure ð Þ *F*½ � Π accounts for the local attributes (all the information quantifiers are evaluated using

For the discrete systems considered here, the logistic map and the delayed logistic map, we find that both *H* and *C* show a correspondence with one of the classic measures of chaoticity, the maximum exponent of Lyapunov, while the local sensitivity of *F* reveals details of the dynamics, invisible to the other quantifiers. The visualization of the location of the dynamics of the system under analysis, in the information planes, allows us to account for (a) the complexity of the system and (b) characterization of different dynamics in different locations of the plane,

enabling the identification of different routes to chaos.

and its maximum value *C* ¼ 0*:*48425. Note also that low entropy values

gives us a sort of "topographical" plane of the dynamics.

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

parameter resolution Δ*r:*

BP-PDF denoted by Π).

**100**

**5. Conclusions**

Felipe Olivares<sup>1</sup> , Lindiane Souza<sup>2</sup> , Walter Legnani<sup>3</sup> and Osvaldo A. Rosso2,4\*

1 Instituto Física, Pontificia Universidad Católica de Valparaíso (PUCV), Valparaíso, Chile

2 Instituto Física, Universidad Federal de Alagoas (UFAL), Maceió, Brazil

3 Signals and Images Processsing Center (CSPSI), Facultad Regional Buenos Aires, Universidad Tecnológica Naciona (UTN), Ciudad Autónoma de Buenos Aires, Argentina

4 Instituto de Medicina Traslacional e Ingeniería Biomedica (IMTIB), Hospital Italiano de Buenos Aires (HIBA), CONICET, Ciudad Autónoma de Buenos Aires, Argentina

\*Address all correspondence to: oarosso@gmail.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

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[2] Sinai YG. On the concept of entropy for a dynamical system. Doklady Akademii Nauk SSSR. 1959;**124**:768-771

[3] Pesin YB. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Mathematical Surveys. 1977;**32**: 55-114

[4] Abarbanel HDI. Analysis of Observed Chaotic Data. New York, USA: Springer-Verlag; 1996

[5] Gray RM. Entropy and Information Theory. Berlin-Heidelberg, Germany: Springer; 1990

[6] Shannon C, Weaver W. The Mathematical Theory of Communication. Champaign, IL: University of Illinois Press; 1949

[7] Brissaud JB. The meaning of entropy. Entropy. 2005;**7**:68-96

[8] Fisher RA. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London. Series A. 1922;**222**: 309-368

[9] Frieden BR. Science from Fisher information: A Unification. Cambridge: Cambridge University Press; 2004

[10] Zografos K, Ferentinos K, Papaioannou T. Discrete approximations to the Csiszár, Renyi, and Fisher measures of information. Canadian Journal of Statistics. 1986;**14**: 355-366

[11] Pardo L, Morales D, Ferentinos K, Zografos K. Discretization problems on generalized entropies and R-

divergences. Kybernetika. 1994;**30**: 445-460

[20] Kowalski A, Martín MT, Plastino A, Rosso AO. Bandt-Pompe approach to the classical-quantum transition.

*DOI: http://dx.doi.org/10.5772/intechopen.88107*

[30] Cabrera JL, De La Rubia FJ.

[32] Cabrera JL, De La Rubia FJ.

[34] Schwarz K. The Archive of

Resonance-like phenomena induced by exponentially correlated parametric noise. Europhysics Letters. 1997;**39**:

[33] Morimoto Y. Hopf bifurcation in the nonlinear recurrence equation xt+1= a xt (1-xt-1). Physics Letters A. 1988;**13**:

Interesting Code. 2011. Available from: http://www.keithschwarz.com/intere sting/code/?dir=factoradic-permutation

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**6**:1683-1690

123-128

179-182

Numerical analysis of transient behavior in discrete random logistic equation with delay. Physics Letters A. 1995;**197**:

[31] Cabrera JL, De La Rubia FJ. Analysis of the behavior of a random nonlinear delay discrete equation. International Journal of Bifurcation and Chaos. 1996;

Physica D: Nonlinear Phenomena. 2007;

[22] Amigó JM. Permutation Complexity in Dynamical Systems. Berlin, Germany:

[23] Rosso OA, Larrondo HA, Martín MT, Plastino A, Fuentes MA. Distinguishing noise from chaos. Physical Review Letters. 2007;**99**:

[24] Olivares F, Plastino A, Rosso OA. Ambiguities in the Bandt and Pompe's methodology for local entropic quantifiers. Physica A: Statistical Mechanics and Its Applications. 2012;

[25] Rosso OA, Olivares F, Plastino A. Noise versus chaos in a causal Fisher-Shannon plane. Papers in Physics. 2015;

[26] Sprott JC. Chaos and Time Series Analysis. Oxford: Oxford University

[27] Hutchinson GE. Circular casual systems in ecology. Annals of New York Academy of Sciences. 1948;**50**:221-246

[28] Pounder JR, Rogers TD. The geometry of chaos: Dynamics of a nonlinear second order difference equation. Bulletin of Mathematical

[29] Aronson DG, Chory MA, Hall GR, McGehee RP. Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study.

Communications in Mathematical

Biology. 1980;**42**:551-597

Physics. 1982;**83**:303-354

**103**

Springer-Verlag; 2010

[21] Amigó JM, Zambrano S, Sanjuán MAF. True and false forbidden patterns in deterministic and random dynamics. Europhysics Letters. 2007;**79**:50001

**233**:21-31

154102

**391**:2518-2526

**7**:070006

Press; 2004

[12] Sánchez-Moreno P, Yáñez R, Dehesa J. Discrete densities and Fisher information. In: Proceedings of the 14th International Conference on Difference Equations and Applications. Istanbul, Turkey: Ugur-Bahçesehir University Press; 2009. pp. 291-298

[13] Olivares F, Plastino A, Rosso OA. Contrasting chaos with noise via local versus global information quantifiers. Physics Letters A. 2012;**376**:1577-1583

[14] López-Ruiz R, Mancini HL, Calbet X. A statistical measure of complexity. Physics Letters A. 1995;**209**:321-326

[15] Lamberti PW, Martín MT, Plastino A, Rosso OA. Intensive entropic nontriviality measure. Physica A: Statistical Mechanics and Its Applications. 2004; **334**:119-131

[16] Martín MT, Plastino A, Rosso OA. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and Its Applications. 2006; **369**:439-462

[17] Bandt C, Pompe B. Permutation entropy: A natural complexity measure for time series. Physical Review Letters. 2002;**88**:174102

[18] Kowalski AM, Martín MT, Plastino A, George Judge G. On extracting probability distribution Information from time series. Entropy. 2012;**14**: 1829-1841

[19] Saco PM, Carpi LC, Figliola A, Serrano E, Rosso AO. Entropy analysis of the dynamics of EL Niño/Southern Oscillation during the Holocene. Physica A: Statistical Mechanics and Its Applications. 2010;**389**:5022-5027

*Informational Time Causal Planes: A Tool for Chaotic Map Dynamic Visualization DOI: http://dx.doi.org/10.5772/intechopen.88107*

[20] Kowalski A, Martín MT, Plastino A, Rosso AO. Bandt-Pompe approach to the classical-quantum transition. Physica D: Nonlinear Phenomena. 2007; **233**:21-31

**References**

55-114

Springer; 1990

[1] Kolmogorov AN. A new metric invariant for transitive dynamical systems and automorphisms in Lebesgue spaces. Doklady Akademii Nauk SSSR. 1959;**119**:861-864

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

divergences. Kybernetika. 1994;**30**:

[12] Sánchez-Moreno P, Yáñez R, Dehesa J. Discrete densities and Fisher information. In: Proceedings of the 14th International Conference on Difference Equations and Applications. Istanbul, Turkey: Ugur-Bahçesehir University

[13] Olivares F, Plastino A, Rosso OA. Contrasting chaos with noise via local versus global information quantifiers. Physics Letters A. 2012;**376**:1577-1583

[14] López-Ruiz R, Mancini HL, Calbet X. A statistical measure of complexity. Physics Letters A. 1995;**209**:321-326

[15] Lamberti PW, Martín MT, Plastino A, Rosso OA. Intensive entropic nontriviality measure. Physica A: Statistical Mechanics and Its Applications. 2004;

[16] Martín MT, Plastino A, Rosso OA. Generalized statistical complexity measures: Geometrical and analytical properties. Physica A: Statistical Mechanics and Its Applications. 2006;

[17] Bandt C, Pompe B. Permutation entropy: A natural complexity measure for time series. Physical Review Letters.

[18] Kowalski AM, Martín MT, Plastino A, George Judge G. On extracting probability distribution Information from time series. Entropy. 2012;**14**:

[19] Saco PM, Carpi LC, Figliola A, Serrano E, Rosso AO. Entropy analysis of the dynamics of EL Niño/Southern Oscillation during the Holocene. Physica

A: Statistical Mechanics and Its Applications. 2010;**389**:5022-5027

Press; 2009. pp. 291-298

445-460

**334**:119-131

**369**:439-462

2002;**88**:174102

1829-1841

[2] Sinai YG. On the concept of entropy for a dynamical system. Doklady Akademii Nauk SSSR. 1959;**124**:768-771

[3] Pesin YB. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Mathematical Surveys. 1977;**32**:

[5] Gray RM. Entropy and Information Theory. Berlin-Heidelberg, Germany:

[4] Abarbanel HDI. Analysis of Observed Chaotic Data. New York,

[6] Shannon C, Weaver W. The Mathematical Theory of

Communication. Champaign, IL: University of Illinois Press; 1949

Entropy. 2005;**7**:68-96

309-368

355-366

**102**

[7] Brissaud JB. The meaning of entropy.

[8] Fisher RA. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London. Series A. 1922;**222**:

[9] Frieden BR. Science from Fisher information: A Unification. Cambridge: Cambridge University Press; 2004

approximations to the Csiszár, Renyi, and Fisher measures of information. Canadian Journal of Statistics. 1986;**14**:

[11] Pardo L, Morales D, Ferentinos K, Zografos K. Discretization problems on

[10] Zografos K, Ferentinos K, Papaioannou T. Discrete

generalized entropies and R-

USA: Springer-Verlag; 1996

[21] Amigó JM, Zambrano S, Sanjuán MAF. True and false forbidden patterns in deterministic and random dynamics. Europhysics Letters. 2007;**79**:50001

[22] Amigó JM. Permutation Complexity in Dynamical Systems. Berlin, Germany: Springer-Verlag; 2010

[23] Rosso OA, Larrondo HA, Martín MT, Plastino A, Fuentes MA. Distinguishing noise from chaos. Physical Review Letters. 2007;**99**: 154102

[24] Olivares F, Plastino A, Rosso OA. Ambiguities in the Bandt and Pompe's methodology for local entropic quantifiers. Physica A: Statistical Mechanics and Its Applications. 2012; **391**:2518-2526

[25] Rosso OA, Olivares F, Plastino A. Noise versus chaos in a causal Fisher-Shannon plane. Papers in Physics. 2015; **7**:070006

[26] Sprott JC. Chaos and Time Series Analysis. Oxford: Oxford University Press; 2004

[27] Hutchinson GE. Circular casual systems in ecology. Annals of New York Academy of Sciences. 1948;**50**:221-246

[28] Pounder JR, Rogers TD. The geometry of chaos: Dynamics of a nonlinear second order difference equation. Bulletin of Mathematical Biology. 1980;**42**:551-597

[29] Aronson DG, Chory MA, Hall GR, McGehee RP. Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study. Communications in Mathematical Physics. 1982;**83**:303-354

[30] Cabrera JL, De La Rubia FJ. Numerical analysis of transient behavior in discrete random logistic equation with delay. Physics Letters A. 1995;**197**: 19-24

[31] Cabrera JL, De La Rubia FJ. Analysis of the behavior of a random nonlinear delay discrete equation. International Journal of Bifurcation and Chaos. 1996; **6**:1683-1690

[32] Cabrera JL, De La Rubia FJ. Resonance-like phenomena induced by exponentially correlated parametric noise. Europhysics Letters. 1997;**39**: 123-128

[33] Morimoto Y. Hopf bifurcation in the nonlinear recurrence equation xt+1= a xt (1-xt-1). Physics Letters A. 1988;**13**: 179-182

[34] Schwarz K. The Archive of Interesting Code. 2011. Available from: http://www.keithschwarz.com/intere sting/code/?dir=factoradic-permutation

**Chapter 6**

**Abstract**

decay estimate, semigroups

**1. Introduction**

**105**

On the Stabilization of Infinite

*El Hassan Zerrik and Abderrahman Ait Aadi*

Illustrations by examples and simulations are also given.

We consider the following semilinear system

various results. In [1], it was shown that the control

and if (3) is replaced by the following assumption

operator such that, for all *ψ* ∈ *H*, we have

*z*ð Þ¼ 0 *z*0*,*

**Keywords:** semilinear systems, output stabilization, feedback controls,

*z t* \_ðÞ¼ *Az t*ðÞþ *v t*ð Þ*Bz t*ð Þ*, t*≥0*,*

weakly stabilizes system (1) provided that *B* be a weakly sequentially continuous

*v t*ðÞ¼�h i *z t*ð Þ*; Bz t*ð Þ *,* (2)

h i *BS t*ð Þ*ψ; S t*ð Þ*ψ* ¼ 0*,* ∀*t*≥ 0 ) *ψ* ¼ 0*,* (3)

where *A* : *D A*ð Þ⊂ *H* ! *H* generates a strongly continuous semigroup of contractions ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on a Hilbert space *H*, endowed with norm and inner product denoted, respectively, by ∥*:*∥ and h i *:; :* , *v*ð Þ*:* ∈*Vad* (the admissible controls set) is a scalar valued control and *B* is a nonlinear operator from *H* to *H* with *B*ð Þ¼ 0 0 so that the origin be an equilibrium state of system (1). The problem of feedback stabilization of distributed system (1) was studied in many works that lead to

(1)

Dimensional Semilinear Systems

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach.

#### **Chapter 6**

## On the Stabilization of Infinite Dimensional Semilinear Systems

*El Hassan Zerrik and Abderrahman Ait Aadi*

#### **Abstract**

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

**Keywords:** semilinear systems, output stabilization, feedback controls, decay estimate, semigroups

#### **1. Introduction**

We consider the following semilinear system

$$\begin{cases} \dot{z}(t) = Az(t) + v(t)Bz(t), & t \ge 0, \\ z(0) = z\_{0\prime} \end{cases} \tag{1}$$

where *A* : *D A*ð Þ⊂ *H* ! *H* generates a strongly continuous semigroup of contractions ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on a Hilbert space *H*, endowed with norm and inner product denoted, respectively, by ∥*:*∥ and h i *:; :* , *v*ð Þ*:* ∈*Vad* (the admissible controls set) is a scalar valued control and *B* is a nonlinear operator from *H* to *H* with *B*ð Þ¼ 0 0 so that the origin be an equilibrium state of system (1). The problem of feedback stabilization of distributed system (1) was studied in many works that lead to various results. In [1], it was shown that the control

$$v(t) = -\langle z(t), Bz(t)\rangle,\tag{2}$$

weakly stabilizes system (1) provided that *B* be a weakly sequentially continuous operator such that, for all *ψ* ∈ *H*, we have

$$
\langle BS(t)\varphi, S(t)\varphi\rangle = 0, \quad \forall t \ge 0 \Rightarrow \varphi = 0,\tag{3}
$$

and if (3) is replaced by the following assumption

$$\int\_{0}^{T} |\langle BS(s)\varphi, S(s)\varphi\rangle| ds \ge \gamma \|\varphi\|^2, \quad \forall \varphi \in H\left(\text{for some } \gamma, T > 0\right),\tag{4}$$

then control (2) strongly stabilizes system (1) [2].

In [3], the authors show that when the resolvent of *A* is compact, *B* self-adjoint and monotone, then strong stabilization of system (1) is proved using bounded controls.

Now, let the output state space *Y* be a Hilbert space with inner product h i *:; : <sup>Y</sup>* and the corresponding norm ∥*:*∥*Y*, and let *C*∈Lð Þ *H; Y* be an output operator.

System (1) is augmented with the output

$$
\omega \upsilon(t) \coloneqq \mathbf{C} z(t). \tag{5}
$$

∥*Cz t*ð Þ∥*<sup>Y</sup>* ! 0*,* as *t* ! ∞*,*

3. exponentially stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global

�*βt*

**Remark 1**. It is clear that exponential stability of (5) ) strong stability of (5) )

The following result provides sufficient conditions for exponential stabilization

Theorem 1.2 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and if the

1. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CAy; <sup>y</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *D A*ð Þ, where *<sup>C</sup>*<sup>∗</sup> is the adjoint operator of *<sup>C</sup>*,

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*y; S t*ð Þ*<sup>y</sup>* <sup>∣</sup>*dt* <sup>≥</sup>*γ*∥*Cy*∥<sup>2</sup>

*v t*ðÞ¼�*ρ*sign *<sup>C</sup>*<sup>∗</sup> ðh *CBz t*ð Þ*; z t*ð ÞÞ

**Proof:** System (1) has a unique mild solution *z t*ð Þ [10] defined on a maximal

*<sup>Y</sup>* <sup>≤</sup> � <sup>2</sup>*<sup>ρ</sup>* <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup>*:*

ð*t* 0

ð*t* 0

*<sup>Y</sup>* ≤ � 2*ρ*

2. ∥*CS t*ð Þ*y*∥*<sup>Y</sup>* ≤*α*∥*Cy*∥*<sup>Y</sup>* and ∥*CBy*∥*<sup>Y</sup>* ≤*β*∥*Cy*∥*Y*, for some *α, β* >0,

∥*z*0∥*,* ∀*t* >0*:*

*Y,* ∀*y*∈ *H,* (6)

*v s*ð Þ*S t*ð Þ � *s Bz s*ð Þ*ds:* (7)

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz*ð Þ*<sup>τ</sup> ; <sup>z</sup>*ð Þ*<sup>τ</sup>* <sup>∣</sup>*dτ:* (8)

∥*Cz t*ð Þ∥*<sup>Y</sup>* ≤ ∥*Cz*0∥*Y:* (9)

∥*Cz t*ð Þ∥*<sup>Y</sup>* ≤ *αe*

and

weak stability of (5).

of the output (5).

condition:

**2.1 Exponential stabilization**

3. there exist *T, γ* > 0 such that

ð*T* 0

exponentially stabilizes the output (5).

*d*

*dt* <sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*Cz*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup>

From hypothesis 1, we deduce

Integrating this inequality, we get

<sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

It follows that

**107**

hold, then there exists *ρ* >0 for which the control

interval 0½ � *; t*max by the variation of constants formula

*z t*ðÞ¼ *S t*ð Þ*z*<sup>0</sup> þ

and there exist *α, β* >0 such that

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

*On the Stabilization of Infinite Dimensional Semilinear Systems*

The output stabilization means that *w t*ðÞ! 0 as *t* ! þ∞ using suitable controls. In the case when *Y* ¼ *H* and *C* ¼ *I*, one obtains the classical stabilization of the state. If Ω be the system evolution domain and *ω*⊂ Ω, when *C* ¼ *χω*, the restriction operator to a subregion *ω* of Ω, one is concerned with the behaviour of the state only in a subregion of the system evolution domain. This is what we call regional stabilization.

The notion of regional stabilization has been largely developed since its closeness to real applications, and the existence of systems which are not stabilizable on the whole domain but stabilizable on some subregion *ω*. Moreover, stabilizing a system on a subregion is cheaper than stabilizing it on the whole domain [4–8]. In [9], the author establishes weak and strong stabilization of (5) for a class of semilinear systems using controls that do not take into account the output operator.

In this paper, we study the output stabilization of semilinear systems by controls that depend on the output operator. Firstly we consider the case of bilinear systems, then we give sufficient conditions to obtain exponential, strong and weak stabilization of the output. Secondly, we consider the case of semilinear systems, and then under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

This paper is organized as follows: In Section 2, we discuss sufficient conditions to achieve exponential, strong and weak stabilization of the output (5) for bilinear systems. In Section 3, we study the output stabilization for a class of semilinear systems. Section 4 is devoted to simulations.

#### **2. Stabilization for bilinear systems**

In this section, we develop sufficient conditions that allow exponential, strong and weak stabilization of the output of bilinear systems. Consider system (1) with *B* : *H* ! *H* is a bounded linear operator and augmented with the output (5).

Definition 1.1 The output (5) is said to be:

1. weakly stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global and satisfies

h i *Cz t*ð Þ*; ψ <sup>Y</sup>* ! 0*,* ∀*ψ* ∈*Y,* as *t* ! ∞*,*

2. strongly stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global and verifies

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

$$\|\mathbf{C}\mathbf{z}(t)\|\_{Y} \to \mathbf{0}, \quad \text{ as } t \to \infty,$$

and

ð*T* 0

simulations are also given.

satisfies

verifies

**106**

systems. Section 4 is devoted to simulations.

**2. Stabilization for bilinear systems**

Definition 1.1 The output (5) is said to be:

controls.

<sup>∣</sup>h i *BS s*ð Þ*ψ; S s*ð Þ*<sup>ψ</sup>* <sup>∣</sup>*ds* <sup>≥</sup>*γ*∥*ψ*∥<sup>2</sup>

System (1) is augmented with the output

then control (2) strongly stabilizes system (1) [2].

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

In [3], the authors show that when the resolvent of *A* is compact, *B* self-adjoint and monotone, then strong stabilization of system (1) is proved using bounded

Now, let the output state space *Y* be a Hilbert space with inner product h i *:; : <sup>Y</sup>* and the corresponding norm ∥*:*∥*Y*, and let *C*∈Lð Þ *H; Y* be an output operator.

The output stabilization means that *w t*ðÞ! 0 as *t* ! þ∞ using suitable controls. In the case when *Y* ¼ *H* and *C* ¼ *I*, one obtains the classical stabilization of the state. If Ω be the system evolution domain and *ω*⊂ Ω, when *C* ¼ *χω*, the restriction operator to a subregion *ω* of Ω, one is concerned with the behaviour of the state only in a subregion of the system evolution domain. This is what we call regional stabilization. The notion of regional stabilization has been largely developed since its closeness to real applications, and the existence of systems which are not stabilizable on the whole domain but stabilizable on some subregion *ω*. Moreover, stabilizing a system on a subregion is cheaper than stabilizing it on the whole domain [4–8]. In [9], the author establishes weak and strong stabilization of (5) for a class of semilinear systems using controls that do not take into account the output operator.

In this paper, we study the output stabilization of semilinear systems by controls that depend on the output operator. Firstly we consider the case of bilinear systems, then we give sufficient conditions to obtain exponential, strong and weak stabilization of the output. Secondly, we consider the case of semilinear systems, and then under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and

This paper is organized as follows: In Section 2, we discuss sufficient conditions to achieve exponential, strong and weak stabilization of the output (5) for bilinear systems. In Section 3, we study the output stabilization for a class of semilinear

In this section, we develop sufficient conditions that allow exponential, strong and weak stabilization of the output of bilinear systems. Consider system (1) with *B* : *H* ! *H* is a bounded linear operator and augmented with the output (5).

1. weakly stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global and

h i *Cz t*ð Þ*; ψ <sup>Y</sup>* ! 0*,* ∀*ψ* ∈*Y,* as *t* ! ∞*,* 2. strongly stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global and

*,* ∀*ψ* ∈ *H* ð Þ for some *γ; T* >0 *,* (4)

*w t*ð Þ ≔ *Cz t*ð Þ*:* (5)

3. exponentially stabilizable, if there exists a control *v*ð Þ*:* ∈*Vad* such that for any initial condition *z*<sup>0</sup> ∈ *H*, the corresponding solution *z t*ð Þ of system (1) is global and there exist *α, β* >0 such that

$$\|\|\mathbf{C}\mathbf{z}(t)\|\|\_{Y} \le ae^{-\beta t} \|\|\mathbf{z}\_0\|\|, \quad \forall t > \mathbf{0}.$$

**Remark 1**. It is clear that exponential stability of (5) ) strong stability of (5) ) weak stability of (5).

#### **2.1 Exponential stabilization**

The following result provides sufficient conditions for exponential stabilization of the output (5).

Theorem 1.2 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and if the condition:


$$\int\_{0}^{T} |\langle \mathbf{C}^\* \text{CBS}(t)y, \mathbf{S}(t)y \rangle| dt \ge \gamma \| \mathbf{C}y \|\_{Y}^2, \forall y \in H,\tag{6}$$

hold, then there exists *ρ* >0 for which the control

$$v(t) = -\rho \operatorname{sign}(\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle)$$

exponentially stabilizes the output (5).

**Proof:** System (1) has a unique mild solution *z t*ð Þ [10] defined on a maximal interval 0½ � *; t*max by the variation of constants formula

$$z(t) = \mathcal{S}(t)z\_0 + \int\_0^t v(s)\mathcal{S}(t-s)Bz(s)ds.\tag{7}$$

From hypothesis 1, we deduce

$$\frac{d}{dt} \|\mathbf{C}\mathbf{z}(t)\|\_{Y}^{2} \le -2\rho \ |\langle \mathbf{C}^{\*}\mathbf{C}\mathbf{B}\mathbf{z}(t), \mathbf{z}(t)\rangle|.$$

Integrating this inequality, we get

$$\left\|\left\|\mathbf{C}\mathbf{z}(t)\right\|\right\|\_{Y}^{2} - \left\|\mathbf{C}\mathbf{z}(\mathbf{0})\right\|\_{Y}^{2} \leq -2\rho \int\_{0}^{t} \left| \left<\mathbf{C}^{\*}\mathbf{C}\mathbf{B}\mathbf{z}(\tau), \mathbf{z}(\tau) \right> \right| d\tau. \tag{8}$$

It follows that

$$\|\mathbf{C}\mathbf{z}(t)\|\_{Y} \le \|\mathbf{C}\mathbf{z}\_{0}\|\_{Y}.\tag{9}$$

For all *z*<sup>0</sup> ∈ *H* and *t*≥ 0, we have

$$
\begin{split}
\langle \mathsf{C}^\* \mathsf{C} \mathsf{B} \mathsf{S}(t) \mathsf{z}\_0, \mathsf{S}(t) \mathsf{z}\_0 \rangle &= \langle \mathsf{C}^\* \mathsf{C} \mathsf{B} \mathsf{z}(t), \mathsf{z}(t) \rangle - \langle \mathsf{C}^\* \mathsf{C} \mathsf{B} \mathsf{z}(t), \mathsf{z}(t) - \mathsf{S}(t) \mathsf{z}\_0 \rangle \\ &+ \langle \mathsf{C}^\* \mathsf{C} \mathsf{B} (\mathsf{S}(t) \mathsf{z}\_0 - \mathsf{z}(t)), \mathsf{S}(t) \mathsf{z}\_0 \rangle.
\end{split}
$$

*w t*ð Þ ≔ *χωz t*ð Þ*,* (13)

<sup>2</sup> � <sup>1</sup> 2*e*<sup>4</sup> � �∥*χωy*∥<sup>2</sup>

*<sup>L</sup>*2ð Þ *<sup>ω</sup>* 6¼ <sup>0</sup>*,*

*<sup>L</sup>*2ð Þ *<sup>ω</sup>* <sup>¼</sup> <sup>0</sup>*,*

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>

*,*

*z t* \_ðÞ¼ *Az t*ð Þþ *f zt* ð Þ ð Þ *, z*ð Þ¼ 0 *z*0*,* (16)

*ds* !*,* as *<sup>t</sup>* ! þ∞*:*

ð Þ Ω *,*

*<sup>ω</sup>* is the adjoint

*<sup>L</sup>*2ð Þ *<sup>ω</sup> :*

*,* (14)

(15)

ð Þ *<sup>ω</sup>* , the restriction operator to *<sup>ω</sup>* and *<sup>χ</sup>* <sup>∗</sup>

<sup>16</sup>*e*<sup>4</sup> , the control

<sup>0</sup> *if* <sup>∥</sup>*χωz t*ð Þ∥<sup>2</sup>

The following result will be used to prove strong stabilization of the output (5). Theorem 1.3 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and

*v t*ðÞ¼� *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ

¼ *O*

In order to make the energy nonincreasing, we consider the control

*v t*ðÞ¼� *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup>

*<sup>Y</sup>* <sup>≤</sup><sup>R</sup> *evt*ð Þ *<sup>C</sup>*<sup>∗</sup> ð Þ h i *CBz t*ð Þ*; z t*ð Þ *:*

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup>

ð*<sup>t</sup>*þ*<sup>T</sup> t*

*<sup>L</sup>*2ð Þ *<sup>ω</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*y*∈*L*<sup>2</sup>

*dx* <sup>¼</sup> <sup>1</sup>

operator of *χω*. Conditions 1 and 3 of Theorem 1.2 hold, indeed: we have

ð2 0 *e* �2*t dt*ð *ω* j j *y* 2

*v t*ðÞ¼ �*<sup>ρ</sup> if* <sup>∥</sup>*χωz t*ð Þ∥<sup>2</sup>

*<sup>ω</sup> χωAy; <sup>y</sup>* � � ¼ �∥*χωy*∥<sup>2</sup>

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*B* : *H* ! *H* is a bounded linear operator. If the conditions:

**Proof:** From hypothesis 1 of Theorem 1.3, we have

*dt* <sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

1 2 *d*

so that the resulting closed-loop system is

2. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CBψ; <sup>ψ</sup>* h i *<sup>B</sup>ψ; <sup>ψ</sup>* <sup>≥</sup> <sup>0</sup>*,* <sup>∀</sup>*<sup>ψ</sup>* <sup>∈</sup> *<sup>H</sup>*, hold, then control

where *χω* : *L*<sup>2</sup>

ð2 0 *χ* ∗ *<sup>ω</sup> χωBe*�*<sup>t</sup>*

**2.2 Strong stabilization**

allows the estimate

<sup>j</sup> *<sup>C</sup>*<sup>∗</sup> <sup>h</sup> *CBS s*ð Þ*z t*ð Þ*; S s*ð Þ*z t*ð Þij*ds* � �<sup>2</sup>

ð*T* 0

**109**

and for *T* ¼ 2, we have

ð Þ! <sup>Ω</sup> *<sup>L</sup>*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

*χ* ∗

*y;e* �*t <sup>y</sup>* � �*dt* <sup>¼</sup>

exponentially stabilizes the output (13).

1. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CAψ; <sup>ψ</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*<sup>ψ</sup>* <sup>∈</sup> *D A*ð Þ,

We conclude that for all 0 <*ρ*< *<sup>e</sup>*4�<sup>1</sup>

Using hypothesis 2 and (9), we have

$$|\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{S}(t) \mathbf{z}\_0, \mathbf{S}(t) \mathbf{z}\_0 \rangle| \le |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle| + 2 \rho a \rho \|\| \mathbf{C}(\mathbf{z}(t) - \mathbf{S}(t) \mathbf{z}\_0) \|\_{Y} \|\mathbf{C} \mathbf{z}\_0 \|\_{Y}.$$

It follows that from (7) and condition 2 that

$$|\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{S}(\mathbf{t}) \mathbf{z}\_0, \mathbf{S}(\mathbf{t}) \mathbf{z}\_0 \rangle| \le |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(\mathbf{t}), \mathbf{z}(\mathbf{t}) \rangle| + 2 \rho \alpha^2 \beta^2 T \|\mathbf{C} \mathbf{z}\_0\|\_Y^2. \tag{10}$$

Integrating (10) over the interval 0½ � *; T* and replacing *z*<sup>0</sup> by *z t*ð Þ and using (6), we deduce that

$$\|\left(\chi - 2\rho\alpha^2\beta^2T^2\right)\|\mathbf{C}z(t)\|\_Y^2 \le \int\_t^{t+T} |\langle \mathbf{C}^\* \cdot \mathbf{C} \mathbf{B}z(s), z(s)\rangle| ds.\tag{11}$$

It follows from the inequality (8) that the sequence ∥*Cz n*ð Þ∥*<sup>Y</sup>* decreases and that for all *n* ∈ N, we have

$$\|\|\mathbf{Cz}(nT)\|\|\_{Y}^{2} - \|\|\mathbf{Cz}((n+1)T)\|\|\_{Y}^{2} \geq 2\rho \int\_{nT}^{(n+1)T} |\langle \mathbf{C}^{\*}\mathbf{C}\mathbf{Bz}(s), \mathbf{z}(s)\rangle| ds.$$

Using (11), we deduce

$$\|\|\mathbf{C}\mathbf{z}(nT)\|\|\_{Y}^{2} - \|\|\mathbf{C}\mathbf{z}((n+1)T)\|\|\_{Y}^{2} \geq 2\rho\left(\chi - 2\rho\alpha^{2}\beta^{2}T^{2}\right) \|\|\mathbf{C}\mathbf{z}(nT)\|\|\_{Y}^{2}.$$

Taking 0< *ρ*< *<sup>γ</sup>* <sup>2</sup>*α*2*β*2*T*2, we get

$$\|\|\mathbf{Cz}(nT)\|\|\_{Y}^{2} \geq 2\rho\left(\mathbf{1} + 2\rho\left(\mathbf{y} - 2\rho a^{2}\beta^{2}T^{2}\right)\right) \|\|\mathbf{Cz}((n+1)T)\|\|\_{Y}^{2}$$

Then

$$\|\|Cz(nT)\|\|\_{Y}^{2} \le \frac{1}{M^{n}}\|\|Cz\_{0}\|\|\_{Y}^{2}.$$

where *<sup>M</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*ρ γ* � <sup>2</sup>*ρα*<sup>2</sup>*β*<sup>2</sup> *T*<sup>2</sup> � � � � > 1. Since ∥*Cz t*ð Þ∥*<sup>Y</sup>* decreases, we deduce that

$$\|\mathbf{C}\mathbf{z}(t)\|\_{Y} \le \sqrt{M}e^{\frac{-\ln\left(M\right)}{2T}}\|\mathbf{z}\_{0}\|, \forall t \ge \mathbf{0},$$

which gives the exponential stability of the output (5). Example 1 On Ω ¼�0*,* 1½, we consider the following system

$$\begin{cases} \frac{\partial \mathbf{z}(\mathbf{x},t)}{\partial t} = Az(\mathbf{x},t) + v(t)\mathbf{z}(\mathbf{x},t) & \boldsymbol{\Omega} \times ]\mathbf{0}, \;+\;\mathbf{\mathcal{Q}}[\\\ \mathbf{z}(\mathbf{x},\mathbf{0}) = \mathbf{z}\_0(\mathbf{x}) & \boldsymbol{\Omega}, \end{cases} \tag{12}$$

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ Ω and *Az* ¼ �*z*. The operator *A* generates a semigroup of contractions on *L*<sup>2</sup> ð Þ <sup>Ω</sup> given by *S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*�*<sup>t</sup> z*0. Let *ω* be a subregion of Ω. System (12) is augmented with the output

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

$$
\omega(t) \coloneqq \chi\_a \boldsymbol{z}(t), \tag{13}
$$

where *χω* : *L*<sup>2</sup> ð Þ! <sup>Ω</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *<sup>ω</sup>* , the restriction operator to *<sup>ω</sup>* and *<sup>χ</sup>* <sup>∗</sup> *<sup>ω</sup>* is the adjoint operator of *χω*. Conditions 1 and 3 of Theorem 1.2 hold, indeed: we have

$$
\langle \chi\_{\alpha}^\* \chi\_{\alpha} A \chi, \mathfrak{y} \rangle = - \| \chi\_{\alpha} \mathfrak{y} \|\_{L^2(\alpha)}^2 \le 0, \quad \forall \mathfrak{y} \in L^2(\Omega),
$$

and for *T* ¼ 2, we have

For all *z*<sup>0</sup> ∈ *H* and *t*≥ 0, we have

Using hypothesis 2 and (9), we have

*<sup>γ</sup>* � <sup>2</sup>*ρα*<sup>2</sup>

*β*2 *<sup>T</sup>*<sup>2</sup> � �∥*Cz t*ð Þ∥<sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*Cz n* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* <sup>∥</sup><sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*Cz n* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* <sup>∥</sup><sup>2</sup>

*<sup>Y</sup>* <sup>≥</sup>2*<sup>ρ</sup>* <sup>1</sup> <sup>þ</sup> <sup>2</sup>*ρ γ* � <sup>2</sup>*ρα*<sup>2</sup>

<sup>∥</sup>*Cz nT* ð Þ∥<sup>2</sup>

<sup>∥</sup>*Cz t*ð Þ∥*<sup>Y</sup>* <sup>≤</sup> ffiffiffiffiffi

which gives the exponential stability of the output (5). Example 1 On Ω ¼�0*,* 1½, we consider the following system

*T*<sup>2</sup> � � � � > 1.

Since ∥*Cz t*ð Þ∥*<sup>Y</sup>* decreases, we deduce that

*<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

ð Þ <sup>Ω</sup> given by *S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*�*<sup>t</sup>*

8 < :

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

augmented with the output

tractions on *L*<sup>2</sup>

**108**

*<sup>Y</sup>* ≤ 1 *<sup>M</sup><sup>n</sup>* <sup>∥</sup>*Cz*0∥<sup>2</sup>

*M* <sup>p</sup> *<sup>e</sup>*

*z x*ð Þ¼ *;* 0 *z*0ð Þ *x* Ω*,*

�ln ð Þ *M* <sup>2</sup>*<sup>T</sup> t*

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *Az x*ð Þþ *; <sup>t</sup> v t*ð Þ*z x*ð Þ *; <sup>t</sup>* <sup>Ω</sup>��0*,* <sup>þ</sup> <sup>∞</sup><sup>½</sup>

ð Þ Ω and *Az* ¼ �*z*. The operator *A* generates a semigroup of con-

*z*0. Let *ω* be a subregion of Ω. System (12) is

<sup>2</sup>*α*2*β*2*T*2, we get

we deduce that

for all *n* ∈ N, we have

<sup>∥</sup>*Cz nT* ð Þ∥<sup>2</sup>

<sup>∥</sup>*Cz nT* ð Þ∥<sup>2</sup>

<sup>∥</sup>*Cz nT* ð Þ∥<sup>2</sup>

where *<sup>M</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>2</sup>*ρ γ* � <sup>2</sup>*ρα*<sup>2</sup>*β*<sup>2</sup>

Using (11), we deduce

Taking 0< *ρ*< *<sup>γ</sup>*

Then

It follows that from (7) and condition 2 that

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> ∣≤∣ *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup> <sup>þ</sup> <sup>2</sup>*ρα*2*β*<sup>2</sup>

*<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ � *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ� *S t*ð Þ*z*<sup>0</sup>

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> ∣≤∣ *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup> <sup>þ</sup> <sup>2</sup>*ραβ*∥*Czt* ð Þ ðÞ� *S t*ð Þ*z*<sup>0</sup> <sup>∥</sup>*Y*∥*Cz*0∥*Y:*

Integrating (10) over the interval 0½ � *; T* and replacing *z*<sup>0</sup> by *z t*ð Þ and using (6),

ð*<sup>t</sup>*þ*<sup>T</sup> t*

It follows from the inequality (8) that the sequence ∥*Cz n*ð Þ∥*<sup>Y</sup>* decreases and that

*<sup>Y</sup>* ≥2*ρ*

ðð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>T</sup> nT*

*<sup>Y</sup>* <sup>≥</sup> <sup>2</sup>*ρ γ* � <sup>2</sup>*ρα*<sup>2</sup>

*β*2 *<sup>T</sup>*<sup>2</sup> � � � � <sup>∥</sup>*Cz n* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* <sup>∥</sup><sup>2</sup>

*β*2 *<sup>T</sup>*<sup>2</sup> � �∥*Cz nT* ð Þ∥<sup>2</sup>

*Y:*

∥*z*0∥*,* ∀*t*≥ 0*,*

*<sup>Y</sup>* ≤

<sup>þ</sup> *<sup>C</sup>*<sup>∗</sup> h i *CB S t* ð Þ ð Þ*z*<sup>0</sup> � *z t*ð Þ *; S t*ð Þ*z*<sup>0</sup> *:*

*T*∥*Cz*0∥<sup>2</sup>

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>*ds:* (11)

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>*ds:*

*Y:*

(12)

*Y:*

*<sup>Y</sup>:* (10)

$$\int\_0^2 \langle \chi\_a^\* \chi\_a B e^{-t} \chi, e^{-t} \chi \rangle dt = \int\_0^2 e^{-2t} dt \int\_{\mathcal{O}} |\mathbf{y}|^2 d\mathbf{x} = \left(\frac{\mathbf{1}}{2} - \frac{\mathbf{1}}{2e^4}\right) \left\| \chi\_a \mathbf{y} \right\|\_{L^2(w)}^2.$$

We conclude that for all 0 <*ρ*< *<sup>e</sup>*4�<sup>1</sup> <sup>16</sup>*e*<sup>4</sup> , the control

$$v(t) = \begin{pmatrix} -\rho & \left. \dot{\mathcal{Y}} \right\|\left\|\chi\_{\alpha}\mathbf{z}(t)\right\|\_{L^{2}(\alpha)}^{2} \neq \mathbf{0},\\ \mathbf{0} & \left. \dot{\mathcal{Y}} \right\|\left\|\chi\_{\alpha}\mathbf{z}(t)\right\|\_{L^{2}(\alpha)}^{2} = \mathbf{0}, \end{pmatrix}$$

exponentially stabilizes the output (13).

#### **2.2 Strong stabilization**

The following result will be used to prove strong stabilization of the output (5). Theorem 1.3 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* : *H* ! *H* is a bounded linear operator. If the conditions:


$$v(t) = -\frac{\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle}{\mathbf{1} + |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle|},\tag{14}$$

allows the estimate

$$O\left(\int\_0^T |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{S}(s) \mathbf{z}(t), \mathbf{S}(s) \mathbf{z}(t) \rangle| ds\right)^2 = O\left(\int\_t^{t+T} \frac{|\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(s), \mathbf{z}(s) \rangle|^2}{1 + |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(s), \mathbf{z}(s) \rangle|} ds\right), \text{ as } t \to +\infty. \tag{15}$$

**Proof:** From hypothesis 1 of Theorem 1.3, we have

$$\frac{1}{2}\frac{d}{dt}\|\mathbf{C}\mathbf{z}(t)\|\_{Y}^{2} \leq \mathfrak{R}\boldsymbol{\varepsilon}(\boldsymbol{\nu}(t)\langle\mathbf{C}^{\*}\,\mathbf{C}\mathbf{B}\mathbf{z}(t),\boldsymbol{z}(t)\rangle).$$

In order to make the energy nonincreasing, we consider the control

$$v(t) = -\frac{\langle \mathbf{C}^\* \text{ CBz}(t), z(t) \rangle}{1 + |\langle \mathbf{C}^\* \text{ CBz}(t), z(t) \rangle|},$$

so that the resulting closed-loop system is

$$\dot{z}(t) = Az(t) + f(z(t)), \ z(0) = z\_0. \tag{16}$$

where

$$f(\boldsymbol{\jmath}) = -\frac{\langle \mathbf{C}^\* \, ^\*\mathbf{C} \mathbf{B} \boldsymbol{\jmath}, \boldsymbol{\jmath} \rangle}{1 + |\langle \mathbf{C}^\* \, ^\*\mathbf{C} \mathbf{B} \boldsymbol{\jmath}, \boldsymbol{\jmath} \rangle|} \text{By, for } \text{ all } \boldsymbol{\jmath} \in H^\*$$

Since *f* is locally Lipschitz, then system (16) has a unique mild solution *z t*ð Þ [10] defined on a maximal interval 0½ � *; tmax* by

$$z(t) = \mathcal{S}(t)z\_0 + \int\_0^t \mathcal{S}(t-s)f(z(s))ds.\tag{17}$$

ð*T* 0

output (5).

ð*T* 0

which achieves the proof.

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*ψ; S t*ð Þ*<sup>ψ</sup>* <sup>∣</sup>*dt* <sup>≥</sup>*γ*∥*Cψ*∥<sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*Cz k* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* <sup>∥</sup><sup>2</sup>

<sup>∥</sup>*Cz kT* ð Þ∥<sup>2</sup>

where *x t*ð Þ is the solution of equation *x*<sup>0</sup>

see that *q s*ð Þ is an increasing function such that

<sup>2</sup>þ*<sup>T</sup>* <sup>1</sup>þ*K*∥*B*∥∥*z*0∥<sup>2</sup> ð Þ � �<sup>2</sup> *:*

*βs* 2

*βs* 2

<sup>2</sup> and *q s*ðÞ¼ *<sup>s</sup>* � ð Þ *<sup>I</sup>* <sup>þ</sup> *<sup>p</sup>* �<sup>1</sup>

**Proof**: Using (19), we deduce

From (15) and (22), we have

where *<sup>β</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup> 2 2*K*∥*B*∥2*T* 3

Taking *sk* <sup>¼</sup> <sup>∥</sup>*Cz kT* ð Þ∥<sup>2</sup>

Since *sk*þ<sup>1</sup> ≤ *sk*, we obtain

Taking *p s*ðÞ¼ *βs*

We obtain �*βx t*ð Þ<sup>2</sup> <sup>≤</sup>*x*<sup>0</sup>

deduce

**111**

<sup>∥</sup>*Cz kT* ð Þ∥<sup>2</sup>

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z t*ð Þ*; S s*ð Þ*z t*ð Þ <sup>∣</sup>*ds*<sup>≤</sup> <sup>2</sup>*K*∥*B*∥<sup>2</sup>

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

� <sup>Ð</sup>*t*þ*<sup>T</sup> t*

bounded linear operator. If the assumptions 1, 2 of Theorem 1.3 and

<sup>∥</sup>*Cz t*ð Þ∥*<sup>Y</sup>* <sup>¼</sup> *<sup>O</sup>* <sup>1</sup>

*T*3

〈*C*<sup>∗</sup> j j *CBz s*ð Þ*;z s*ð Þ〉 <sup>2</sup> <sup>1</sup>þ<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*;z s*ð Þ <sup>∣</sup> *ds* � �<sup>1</sup>

The following result gives sufficient conditions for strong stabilization of the

Theorem 1.4 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* is a

holds, then control (14) strongly stabilizes the output (5) with decay estimate

ffiffi *t* p

> ð*k T*ð Þ <sup>þ</sup><sup>1</sup> *kT*

*<sup>Y</sup>*, the inequality (24) can be written as

*<sup>k</sup>* þ *sk*þ<sup>1</sup> ≤*sk,* ∀*k*≥0*:*

*<sup>k</sup>*þ<sup>1</sup> þ *sk*þ<sup>1</sup> ≤ *sk,* ∀*k*≥ 0*:*

*sk* ≤ *x k*ð Þ*, k*≥ 0*,*

Since *x k*ð Þ≥*sk* and *x t*ð Þ decreases give *x t*ð Þ≥ 0, ∀*t*≥0. Furthermore, it is easy to

0≤*q s*ð Þ≤*p s*ð Þ*,* ∀*s* ≥0*:*

*x t*ðÞ¼ *O t*�<sup>1</sup> � �*, as t* ! þ∞*:*

ð Þ*t* ≤0, which implies that

*<sup>Y</sup>* ≥2

*<sup>Y</sup>* � <sup>∥</sup>*Cz k* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup>* <sup>∥</sup><sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>T</sup>*<sup>ð</sup> <sup>1</sup> <sup>þ</sup> *<sup>K</sup>*∥*B*∥∥*z*0∥<sup>2</sup> � � � �

*Y,* ∀*ψ* ∈ *H, for some T* ð Þ *; γ* >0 *,* (22)

� �*,* as *<sup>t</sup>* ! þ∞*:* (23)

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz t*ð Þ*; z t*ð Þ <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup>

*<sup>Y</sup>* <sup>≥</sup>*β*∥*Cz kT* ð Þ∥<sup>4</sup>

ð Þ*s* in Lemma 3.3, page 531 in [11], we

ð Þþ*t qxt* ð Þ¼ ð Þ 0*, x*ð Þ¼ 0 *s*0.

*dt, k*≥0*:*

*Y,* (24)

2 *,*

Because of the contractions of the semigroup, we have

$$\frac{d}{dt} \|\mathbf{z}(t)\|^2 \le -2 \frac{\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle \langle \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle}{\mathbf{1} + |\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle|}.$$

Integrating this inequality, we get

$$\|\|z(t)\|\|^2 - \|z(\mathbf{0})\|\|^2 \le -2 \int\_0^t \frac{\langle \mathbf{C}^\* \text{CBz}(s), z(s) \rangle \langle \mathbf{B}z(s), z(s) \rangle}{\mathbf{1} + |\langle \mathbf{C}^\* \text{CBz}(s), z(s) \rangle|} ds.$$

It follows that

$$\|\|z(t)\|\| \le \|z\_0\|\|.\tag{18}$$

From hypothesis 1 of Theorem 1.3, we have

$$\frac{d}{dt} \|\mathbf{C}\mathbf{z}(t)\|\_{Y}^{2} \leq -2\frac{\left|\langle\mathbf{C}^{\*}\,\mathbf{C}\mathbf{B}\mathbf{z}(t),\mathbf{z}(t)\rangle\right|^{2}}{\mathbf{1}+\left|\langle\mathbf{C}^{\*}\,\mathbf{C}\mathbf{B}\mathbf{z}(t),\mathbf{z}(t)\rangle\right|}.$$

We deduce

$$\left\|\mathrm{Cz}(t)\right\|\_{Y}^{2} - \left\|\mathrm{Cz}(\mathbf{0})\right\|\_{Y}^{2} \leq -2\int\_{0}^{t} \frac{\left|\langle\mathrm{C}^{\*}\mathrm{CBz}(s), \mathrm{z}(s)\rangle\right|^{2}}{\mathbf{1} + \left|\langle\mathrm{C}^{\*}\mathrm{CBz}(s), \mathrm{z}(s)\rangle\right|} ds.\tag{19}$$

Using (17) and Schwartz inequality, we get

$$\|\|z(t) - \mathcal{S}(t)z\_0\| \le \|B\| \|\|z\_0\| \left( T \int\_0^t \frac{|\langle \mathcal{C}^\* \mathcal{C} \mathcal{B} \mathbf{z}(s), z(s) \rangle|^2}{\mathbf{1} + |\langle \mathcal{C}^\* \mathcal{C} \mathcal{B} \mathbf{z}(s), z(s) \rangle|} ds \right)^{\frac{1}{2}}, \ \forall t \in [0, T]. \tag{20}$$

Since *B* is bounded and *C* continuous, we have

$$|\langle \mathbf{C}^\* \text{CBS}(\mathfrak{s}) \mathbf{z}\_0, \mathbf{S}(\mathfrak{s}) \mathbf{z}\_0 \rangle| \le 2K \|\mathbf{B}\| \|\mathbf{z}(\mathfrak{s}) - \mathbf{S}(\mathfrak{s}) \mathbf{z}\_0\| \|\mathbf{z}\_0\| + |\langle \mathbf{C}^\* \text{CBS}(\mathfrak{s}), \mathbf{z}(\mathfrak{s}) \rangle|,\tag{21}$$

where *K* is a positive constant.

Replacing *z*<sup>0</sup> by *z t*ð Þ in (20) and (21), we get

$$\begin{split} |\langle \mathsf{C}^\* \,^\* \mathrm{CBS}(s) z(t), \mathrm{S}(s) z(t) \rangle| &\leq 2K \|B\|^2 \|\mathrm{z}\_0\|^2 \Big( T \int\_t^{t+T} \frac{|\langle \mathrm{C}^\* \,^\* \mathrm{CBr}(s), x(s) \rangle|^2}{1 + |\langle \mathrm{C}^\* \,^\* \mathrm{CBr}(s), x(s) \rangle|^2} ds \Big)^{\frac{1}{2}} \\ &+ |\langle \mathrm{C}^\* \,^\* \mathrm{CBr}(t+s), x(t+s) \rangle|, \qquad \forall t \geq s \geq 0. \end{split}$$

Integrating this relation over 0½ � *; T* and using Cauchy-Schwartz, we deduce

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

$$\begin{aligned} \int\_0^T |\langle \mathbf{C}^\* \operatorname{CBS}(s) \mathbf{z}(t), \mathbf{S}(s) \mathbf{z}(t) \rangle| ds &\leq \left( 2K \|\|B\|\|^2 T^{\frac{3}{2}} + T(\left( \mathbf{1} + K \|\|B\|\| \|\mathbf{z}\_0\|\|^2 \right) \right)^{\frac{1}{2}} \\ &\times \left( \int\_t^{t+T} \frac{|\langle \mathbf{C}^\* \operatorname{CBS}(s) \mathbf{z}(s) \rangle|^2}{1 + |\langle \mathbf{C}^\* \operatorname{CBS}(s) \mathbf{z}(s) \rangle|} ds \right)^{\frac{1}{2}}, \end{aligned}$$

which achieves the proof.

where

*f y* ð Þ¼� *<sup>C</sup>*<sup>∗</sup> h i *CBy; <sup>y</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*z t*ðÞ¼ *S t*ð Þ*z*<sup>0</sup> þ

Because of the contractions of the semigroup, we have

defined on a maximal interval 0½ � *; tmax* by

*d*

Integrating this inequality, we get

It follows that

We deduce

**110**

<sup>∥</sup>*z t*ð Þ∥<sup>2</sup> � <sup>∥</sup>*z*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup> <sup>≤</sup> � <sup>2</sup>

From hypothesis 1 of Theorem 1.3, we have

*dt* <sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*Cz*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup>

ð*t* 0

Using (17) and Schwartz inequality, we get

Since *B* is bounded and *C* continuous, we have

Replacing *z*<sup>0</sup> by *z t*ð Þ in (20) and (21), we get

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z t*ð Þ*; S s*ð Þ*z t*ð Þ ∣ ≤2*K*∥*B*∥<sup>2</sup>

*d*

<sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

∥*z t*ðÞ� *S t*ð Þ*z*0∥≤∥*B*∥∥*z*0∥ *T*

where *K* is a positive constant.

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBy; <sup>y</sup>* <sup>∣</sup>

Since *f* is locally Lipschitz, then system (16) has a unique mild solution *z t*ð Þ [10]

ð*t* 0

*dt* <sup>∥</sup>*z t*ð Þ∥<sup>2</sup> <sup>≤</sup> � <sup>2</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ h i *Bz t*ð Þ*; z t*ð Þ

*<sup>Y</sup>* <sup>≤</sup> � <sup>2</sup> *<sup>C</sup>*<sup>∗</sup> j j h i *CBz t*ð Þ*; z t*ð Þ <sup>2</sup>

ð*t* 0

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>

> ∥*z*0∥<sup>2</sup> *T* Ð*<sup>t</sup>*þ*<sup>T</sup> t*

<sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ <sup>þ</sup> *<sup>s</sup> ; z t*ð Þ <sup>þ</sup> *<sup>s</sup>* <sup>∣</sup>*,* <sup>∀</sup>*<sup>t</sup>* <sup>≥</sup>*s*≥0*:*

!<sup>1</sup>

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z*0*; S s*ð Þ*z*<sup>0</sup> ∣ ≤ <sup>2</sup>*K*∥*B*∥∥*z s*ðÞ� *S s*ð Þ*z*0∥∥*z*0<sup>∥</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>*,* (21)

Integrating this relation over 0½ � *; T* and using Cauchy-Schwartz, we deduce

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup>

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup>

*ds*

2

〈*C*<sup>∗</sup> j j *CBz s*ð Þ*;z s*ð Þ〉 <sup>2</sup> <sup>1</sup>þ<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*;z s*ð Þ <sup>∣</sup> *ds* � �<sup>1</sup>

ð*t* 0

*<sup>Y</sup>* ≤ � 2

*By,*for all *y*∈ *H*

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup> *:*

*<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ h i *Bz s*ð Þ*; z s*ð Þ <sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz s*ð Þ*; z s*ð Þ <sup>∣</sup> *ds:*

∥*z t*ð Þ∥≤∥*z*0∥*:* (18)

*:*

*ds:* (19)

*,* ∀*t*∈½ � 0*; T :* (20)

2

*S t*ð Þ � *s f zs* ð Þ ð Þ *ds:* (17)

The following result gives sufficient conditions for strong stabilization of the output (5).

Theorem 1.4 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* is a bounded linear operator. If the assumptions 1, 2 of Theorem 1.3 and

$$\int\_{0}^{T} |\langle \mathbb{C}^\* \text{CBS}(t)\varphi, \mathbb{S}(t)\varphi \rangle| dt \ge \gamma \|\mathbb{C}\varphi\|\_{Y}^2, \quad \forall \varphi \in H, \text{ (for some } T, \gamma > 0), \tag{22}$$

holds, then control (14) strongly stabilizes the output (5) with decay estimate

$$\|\mathbb{C}\mathbf{z}(t)\|\_{Y} = O\left(\frac{1}{\sqrt{t}}\right), \quad \text{as} \quad t \to +\infty. \tag{23}$$

**Proof**: Using (19), we deduce

$$\|\mathsf{Cz}(kT)\|\_{Y}^{2} - \|\mathsf{Cz}((k+1)T)\|\_{Y}^{2} \geq 2\int\_{kT}^{k(T+1)} \frac{|\langle \mathsf{C^{\*}}\,\mathsf{CBz}(t), \mathsf{z}(t)\rangle|^{2}}{\mathbf{1} + |\langle \mathsf{C^{\*}}\,\mathsf{CBz}(t), \mathsf{z}(t)\rangle|} dt, \ k \geq 0.1$$

From (15) and (22), we have

$$\|\|\mathbf{Cz}(kT)\|\|\_{Y}^{2} - \|\|\mathbf{Cz}((k+1)T)\|\|\_{Y}^{2} \geq \beta \|\|\mathbf{Cz}(kT)\|\|\_{Y}^{4} \tag{24}$$

where *<sup>β</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>2</sup> 2 2*K*∥*B*∥2*T* 3 <sup>2</sup>þ*<sup>T</sup>* <sup>1</sup>þ*K*∥*B*∥∥*z*0∥<sup>2</sup> ð Þ � �<sup>2</sup> *:*

Taking *sk* <sup>¼</sup> <sup>∥</sup>*Cz kT* ð Þ∥<sup>2</sup> *<sup>Y</sup>*, the inequality (24) can be written as

$$
\beta \mathfrak{s}\_k^2 + s\_{k+1} \le s\_k, \quad \forall k \ge 0.
$$

Since *sk*þ<sup>1</sup> ≤ *sk*, we obtain

$$
\beta s\_{k+1}^2 + s\_{k+1} \le s\_k, \quad \forall k \ge 0.
$$

Taking *p s*ðÞ¼ *βs* <sup>2</sup> and *q s*ðÞ¼ *<sup>s</sup>* � ð Þ *<sup>I</sup>* <sup>þ</sup> *<sup>p</sup>* �<sup>1</sup> ð Þ*s* in Lemma 3.3, page 531 in [11], we deduce

$$s\_k \le \varkappa(k), \quad k \ge 0,$$

where *x t*ð Þ is the solution of equation *x*<sup>0</sup> ð Þþ*t qxt* ð Þ¼ ð Þ 0*, x*ð Þ¼ 0 *s*0.

Since *x k*ð Þ≥*sk* and *x t*ð Þ decreases give *x t*ð Þ≥ 0, ∀*t*≥0. Furthermore, it is easy to see that *q s*ð Þ is an increasing function such that

$$0 \le q(s) \le p(s), \forall s \ge 0.$$

We obtain �*βx t*ð Þ<sup>2</sup> <sup>≤</sup>*x*<sup>0</sup> ð Þ*t* ≤0, which implies that

$$\varkappa(t) = O(t^{-1}), \quad \text{as} \ t \to +\infty.$$

Finally the inequality *sk* ≤*x k*ð Þ, together with the fact that ∥*Cz t*ð Þ∥*<sup>Y</sup>* decreases, we deduce the estimate (23).

Example 2 Let us consider a system defined on Ω ¼�0*,* 1½ by

$$\begin{cases} \frac{\partial z(\mathbf{x},t)}{\partial t} = Az(\mathbf{x},t) + v(t)a(\mathbf{x})z(\mathbf{x},t) & \boldsymbol{\Omega} \times ]\mathbf{0}, \;+\;\boldsymbol{\infty}[\\\\ z(\mathbf{x},0) = z\_0(\mathbf{x}) & \boldsymbol{\Omega} \\\\ z(\mathbf{0},t) = z(\mathbf{1},t) = \mathbf{0} & t > \mathbf{0}, \end{cases} \tag{25}$$

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ <sup>Ω</sup> , *Az* ¼ �*z*, and *<sup>a</sup>*∈*L*<sup>∞</sup>ð�0*,* <sup>1</sup>½Þ such that *a x*ð Þ≥0 a.e on �0*,* <sup>1</sup><sup>½</sup> and *a x*ð Þ≥*c*>0 on subregion *<sup>ω</sup>* of <sup>Ω</sup> and *<sup>v</sup>*ð Þ*:* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup>ð Þ <sup>0</sup>*;* <sup>þ</sup><sup>∞</sup> the control function. System (25) is augmented with the output

$$
\omega w(\mathbf{t}) = \chi\_o \mathbf{z}(\mathbf{t}).\tag{26}
$$

Γ *t<sup>ϕ</sup>*ð Þ *<sup>n</sup>*

ð*<sup>ϕ</sup>*ð Þþ *<sup>n</sup> <sup>t</sup> ϕ*ð Þ *n*

<sup>∣</sup> *<sup>C</sup>*∗*CBS s*ð Þ<sup>Γ</sup> *<sup>t</sup>ϕ*ð Þ *<sup>n</sup>*

Hence, by the dominated convergence Theorem, we have

ð*t* 0

Using condition 3 of Theorem 1.5, we deduce that

*C*Γ *tϕ*ð Þ *<sup>n</sup>*

*<sup>∂</sup><sup>t</sup>* ¼ � *<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

ð Þ *S t*ð Þ*z*<sup>0</sup> ð Þ¼ *x*

*∂x*

h i *C*Γð Þ *tn z*0*; φ* ! 0 as *n* ! þ∞ and hence

*<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

*<sup>∂</sup><sup>x</sup>* with domain

8 >>>><

>>>>:

where *Az* ¼ � *<sup>∂</sup><sup>z</sup>*

*D A*ð Þ¼ *<sup>z</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup>

**113**

� �*z*0*; S t*ð Þ<sup>Γ</sup> *<sup>t</sup><sup>ϕ</sup>*ð Þ *<sup>n</sup>*

Since *B* is compact and *C* continuous, we have

*On the Stabilization of Infinite Dimensional Semilinear Systems*

lim*n*!þ<sup>∞</sup> *<sup>C</sup>*<sup>∗</sup>*CBS t*ð Þ<sup>Γ</sup> *<sup>t</sup><sup>ϕ</sup>*ð Þ *<sup>n</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

Λ*n*ð Þ*t* ≔

ð*t* 0

It follows that ∀*t*≥ 0, Λ*n*ðÞ!*t* 0 as *n* ! þ∞.

For all *n*≥, we set

Using (15), we get

We conclude that

such that *C*Γ *tϕ*ð Þ *<sup>n</sup>*

lim*<sup>n</sup>*!þ<sup>∞</sup>

� �*z*<sup>0</sup> ⇀ *<sup>ψ</sup>, as n* ! <sup>∞</sup>*:*

� �*z*<sup>0</sup> � � <sup>¼</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*ψ; S t*ð Þ*<sup>ψ</sup> :*

*<sup>C</sup>*<sup>∗</sup> j j h i *CB*Γð Þ*<sup>s</sup> <sup>z</sup>*0*;* <sup>Γ</sup>ð Þ*<sup>s</sup> <sup>z</sup>*<sup>0</sup>

<sup>1</sup> <sup>þ</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CB*Γð Þ*<sup>s</sup> <sup>z</sup>*0*;* <sup>Γ</sup>ð Þ*<sup>s</sup> <sup>z</sup>*<sup>0</sup> <sup>∣</sup>

� �*z*0*; S s*ð Þ<sup>Γ</sup> *<sup>t</sup>ϕ*ð Þ *<sup>n</sup>*

� �∣*ds* <sup>¼</sup> <sup>0</sup>*:*

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*ψ; S s*ð Þ*<sup>ψ</sup>* <sup>∣</sup>*ds* <sup>¼</sup> <sup>0</sup>*:*

*<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*ψ; S s*ð Þ*<sup>ψ</sup>* <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*s*∈½ � <sup>0</sup>*; <sup>t</sup> :*

On the other hand, it is clear that (27) holds for each subsequence *tϕ*ð Þ *<sup>n</sup>*

Example 3 Consider a system defined in Ω ¼�0*,* þ ∞½, and described by

*z x*ð Þ¼ *;* 0 *z*0ð Þ *x x*∈ Ω *z*ð Þ¼ 0*; t z*ð Þ¼ ∞*; t* 0 *t*> 0*,*

ð Þj <sup>Ω</sup> *<sup>z</sup>*ð Þ¼ <sup>0</sup> <sup>0</sup>*; z x*ð Þ! <sup>0</sup> *as x* ! þ<sup>∞</sup> � � and *Bz*ðÞ¼ *:* <sup>Ð</sup> <sup>1</sup>

the control operator. The operator *A* generates a semigroup of contractions

�

Let *ω* ¼�0*,* 1½ be a subregion of Ω and system (28) is augmented with the output

*C*Γð Þ*t z*<sup>0</sup> ⇀ 0*,* as *t* ! þ∞*:*

� �*z*<sup>0</sup> weakly converges in *Y*. This implies that ∀*φ*∈*Y*, we have

2

� �*z*<sup>0</sup>

� �*z*<sup>0</sup> ⇀ <sup>0</sup>*,* as *<sup>n</sup>* ! þ∞*:* (27)

þ *v t*ð Þ*Bz x*ð Þ *; t x*∈ Ω*, t*>0

*z*0ð Þ *x* � *t* if *x*≥*t* 0 if *x*<*t:* � � of ð Þ *tn*

(28)

<sup>0</sup> *z x*ð Þ*dx*ð Þ*:* is

*ds:*

The operator *A* generates a semigroup of contractions on *L*<sup>2</sup> ð Þ Ω given by *S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*�*<sup>t</sup> z*0. For *z*<sup>0</sup> ∈*L*<sup>2</sup> ð Þ Ω and *T* ¼ 2, we obtain

$$\int\_0^2 \langle \chi\_{\, \alpha}^\* \chi\_{\, \alpha} B S(t) z\_0, S(t) z\_0 \rangle dt = \int\_0^2 e^{-2t} dt \int\_{\alpha} a(\infty) |z\_0|^2 d\mathfrak{x} \ge \beta \| |\chi\_{\, \alpha} z\_0| \|\_{L^2(\alpha)}^2$$

with *β* ¼ *c* Ð 2 <sup>0</sup> *e*�2*<sup>t</sup> dt*>0.

Applying Theorem 1.4, we conclude that the control

$$\nu(t) = -\frac{\int\_{\alpha} a(\varkappa) |z(\varkappa, t)|^2 d\varkappa}{1 + \int\_{\alpha} a(\varkappa) |z(\varkappa, t)|^2 d\varkappa}$$

strongly stabilizes the output (26) with decay estimate

$$\|\chi\_{\alpha}\mathbf{z}(t)\|\_{L^{2}(\alpha)} = O\left(\frac{1}{\sqrt{t}}\right), \quad \text{as} \quad t \to +\infty.$$

#### **2.3 Weak stabilization**

The following result provides sufficient conditions for weak stabilization of the output (5).

Theorem 1.5 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* is a compact operator. If the conditions:


**Proof:** Let us consider the nonlinear semigroup Γð Þ*t z*<sup>0</sup> ≔ *z t*ð Þ and let ð Þ *tn* be a sequence of real numbers such that *tn* ! þ∞ as *n* ! þ∞.

From (18), Γð Þ *tn z*<sup>0</sup> is bounded in *H*, then there exists a subsequence *t<sup>ϕ</sup>*ð Þ *<sup>n</sup>* � � of ð Þ *tn* such that

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

$$
\Gamma \left( t\_{\phi(n)} \right) z\_0 \rightharpoonup \psi, \quad \text{as } n \to \infty.
$$

Since *B* is compact and *C* continuous, we have

$$\lim\_{n \to +\infty} \left< \mathcal{C}^\* \text{CBS}(t) \Gamma \left( t\_{\phi(n)} \right) z\_0, \mathcal{S}(t) \Gamma \left( t\_{\phi(n)} \right) z\_0 \right> = \left< \mathcal{C}^\* \text{CBS}(t) \boldsymbol{\varphi}, \mathcal{S}(t) \boldsymbol{\varphi} \right>.$$

For all *n*≥, we set

Finally the inequality *sk* ≤*x k*ð Þ, together with the fact that ∥*Cz t*ð Þ∥*<sup>Y</sup>* decreases,

*<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *Az x*ð Þþ *; <sup>t</sup> v t*ð Þ*a x*ð Þ*z x*ð Þ *; <sup>t</sup>* <sup>Ω</sup>��0*,* <sup>þ</sup> <sup>∞</sup><sup>½</sup>

ð Þ <sup>Ω</sup> , *Az* ¼ �*z*, and *<sup>a</sup>*∈*L*<sup>∞</sup>ð�0*,* <sup>1</sup>½Þ such that *a x*ð Þ≥0 a.e on �0*,* <sup>1</sup><sup>½</sup> and

*a x*ð Þj j *z*<sup>0</sup> 2

*dx*

*,* as *t* ! þ∞*:*

*dx*

*w t*ðÞ¼ *χωz t*ð Þ*:* (26)

ð Þ Ω given by

*<sup>L</sup>*2ð Þ *<sup>ω</sup> ,*

� � of ð Þ *tn*

*dx*≥*β*∥*χωz*0∥<sup>2</sup>

(25)

Example 2 Let us consider a system defined on Ω ¼�0*,* 1½ by

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

The operator *A* generates a semigroup of contractions on *L*<sup>2</sup>

*z x*ð Þ¼ *;* 0 *z*0ð Þ *x* Ω

*z*ð Þ¼ 0*; t z*ð Þ¼ 1*; t* 0 *t*>0*,*

ð Þ Ω and *T* ¼ 2, we obtain

ð2 0 *e* �2*t dt* ð *ω*

Ð

<sup>1</sup> <sup>þ</sup> <sup>Ð</sup>

*<sup>ω</sup>a x*ð Þj j *z x*ð Þ *; <sup>t</sup>* <sup>2</sup>

ffiffi *t* p � �

The following result provides sufficient conditions for weak stabilization of the

Theorem 1.5 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* is a

3. *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*ψ; S t*ð Þ*<sup>ψ</sup>* <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*t*≥<sup>0</sup> ) *<sup>C</sup><sup>ψ</sup>* <sup>¼</sup> 0 hold, then control (14) weakly

**Proof:** Let us consider the nonlinear semigroup Γð Þ*t z*<sup>0</sup> ≔ *z t*ð Þ and let ð Þ *tn* be a

From (18), Γð Þ *tn z*<sup>0</sup> is bounded in *H*, then there exists a subsequence *t<sup>ϕ</sup>*ð Þ *<sup>n</sup>*

*<sup>ω</sup>a x*ð Þj j *z x*ð Þ *; <sup>t</sup>* <sup>2</sup>

*a x*ð Þ≥*c*>0 on subregion *<sup>ω</sup>* of <sup>Ω</sup> and *<sup>v</sup>*ð Þ*:* <sup>∈</sup> *<sup>L</sup>*<sup>∞</sup>ð Þ <sup>0</sup>*;* <sup>þ</sup><sup>∞</sup> the control function. System

we deduce the estimate (23).

8 >>>><

>>>>:

(25) is augmented with the output

*z*0. For *z*<sup>0</sup> ∈*L*<sup>2</sup>

Ð 2 <sup>0</sup> *e*�2*<sup>t</sup>*

*<sup>ω</sup> χωBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> � �*dt* <sup>¼</sup>

*dt*>0.

Applying Theorem 1.4, we conclude that the control

*v t*ðÞ¼�

strongly stabilizes the output (26) with decay estimate

<sup>∥</sup>*χωz t*ð Þ∥*L*2ð Þ *<sup>ω</sup>* <sup>¼</sup> *<sup>O</sup>* <sup>1</sup>

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

*S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>e</sup>*�*<sup>t</sup>*

ð2 0 *χ* ∗

with *β* ¼ *c*

**2.3 Weak stabilization**

compact operator. If the conditions:

stabilizes the output (5).

1. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CAψ; <sup>ψ</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*<sup>ψ</sup>* <sup>∈</sup> *D A*ð Þ,

2. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CBψ; <sup>ψ</sup>* h i *<sup>B</sup>ψ; <sup>ψ</sup>* <sup>≥</sup> <sup>0</sup>*,* <sup>∀</sup>*<sup>ψ</sup>* <sup>∈</sup> *<sup>H</sup>*,

sequence of real numbers such that *tn* ! þ∞ as *n* ! þ∞.

output (5).

such that

**112**

*<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

$$\Lambda\_{\mathfrak{n}}(t) \coloneqq \int\_{\phi(n)}^{\phi(n)+t} \frac{\left| \langle \mathbf{C}^\* \,^\*\mathbf{C} \mathbf{B} \Gamma(s) \mathbf{z}\_0, \Gamma(s) \mathbf{z}\_0 \rangle \right|^2}{\mathbf{1} + \left| \langle \mathbf{C}^\* \,^\*\mathbf{C} \mathbf{B} \Gamma(s) \mathbf{z}\_0, \Gamma(s) \mathbf{z}\_0 \rangle \right|} ds.$$

It follows that ∀*t*≥ 0, Λ*n*ðÞ!*t* 0 as *n* ! þ∞. Using (15), we get

$$\lim\_{n \to +\infty} \int\_0^t | \langle \mathcal{C}^\* \mathcal{C} \mathcal{S} \mathcal{S}(s) \Gamma (t\_{\phi(n)}) z\_0, \mathcal{S}(s) \Gamma (t\_{\phi(n)}) z\_0 \rangle | ds = \mathbf{0}.$$

Hence, by the dominated convergence Theorem, we have

$$\int\_0^t |\langle C^\*CBS(s)\boldsymbol{\nu}, S(s)\boldsymbol{\nu}\rangle| ds = \mathbf{0}.$$

We conclude that

$$
\langle \mathsf{C}^\* \mathsf{CBS}(s)\mu, \mathsf{S}(s)\mu \rangle = \mathsf{0}, \quad \forall s \in [\mathsf{0}, t].
$$

Using condition 3 of Theorem 1.5, we deduce that

$$\text{CT}\left(t\_{\phi(n)}\right)z\_0 \rightharpoonup \mathbf{0}, \quad \text{as} \quad n \to +\infty. \tag{27}$$

On the other hand, it is clear that (27) holds for each subsequence *tϕ*ð Þ *<sup>n</sup>* � � of ð Þ *tn* such that *C*Γ *tϕ*ð Þ *<sup>n</sup>* � �*z*<sup>0</sup> weakly converges in *Y*. This implies that ∀*φ*∈*Y*, we have h i *C*Γð Þ *tn z*0*; φ* ! 0 as *n* ! þ∞ and hence

$$\text{CT}(t)\text{z}\_0 \rightharpoonup \text{0}, \quad \text{as} \quad t \to +\infty.$$

Example 3 Consider a system defined in Ω ¼�0*,* þ ∞½, and described by

$$\begin{cases} \frac{\partial z(\mathbf{x},t)}{\partial t} = -\frac{\partial z(\mathbf{x},t)}{\partial \mathbf{x}} + v(t)Bz(\mathbf{x},t) & \mathbf{x} \in \Omega, \ t > 0 \\\\ z(\mathbf{x},0) = z\_0(\mathbf{x}) & \mathbf{x} \in \Omega \\\\ z(0,t) = z(\infty,t) = 0 & t > 0, \end{cases} \tag{28}$$

where *Az* ¼ � *<sup>∂</sup><sup>z</sup> <sup>∂</sup><sup>x</sup>* with domain

*D A*ð Þ¼ *<sup>z</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup> ð Þj <sup>Ω</sup> *<sup>z</sup>*ð Þ¼ <sup>0</sup> <sup>0</sup>*; z x*ð Þ! <sup>0</sup> *as x* ! þ<sup>∞</sup> � � and *Bz*ðÞ¼ *:* <sup>Ð</sup> <sup>1</sup> <sup>0</sup> *z x*ð Þ*dx*ð Þ*:* is the control operator. The operator *A* generates a semigroup of contractions

$$(\mathcal{S}(t)z\_0)(x) = \begin{cases} z\_0(x-t) & \text{if } x \ge t \\ 0 & \text{if } x < t. \end{cases}$$

Let *ω* ¼�0*,* 1½ be a subregion of Ω and system (28) is augmented with the output

$$
\omega(\mathbf{t}) = \chi\_o \mathbf{z}(\mathbf{t}).\tag{29}
$$

*v t*ðÞ¼ � *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ

**Proof:** Since ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is a semigroup of contractions, we have

8 < :

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*d*

exponentially stabilizes the output (5).

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

For all *z*<sup>0</sup> ∈ *H* and *t*≥ 0, we have

where *α* is a positive constant.

Using (33), we deduce

∥*z*0∥ such that

we obtain

we get

**115**

Ð *T*

semigroup ð Þ *U t*ð Þ *<sup>t</sup>*≥<sup>0</sup>, we deduce that

� <sup>Ð</sup>*<sup>t</sup>*þ*<sup>T</sup>*

<sup>þ</sup> *<sup>T</sup>*<sup>1</sup>

Ð *T*

<sup>∥</sup>*z t*ð Þ∥<sup>2</sup> *,* if *z t*ð Þ*=*<sup>¼</sup> <sup>0</sup>*,*

∥*z t*ð Þ∥≤∥*z*0∥*:* (33)

(32)

0*,* if *z t*ðÞ¼ 0*,*

*dt* <sup>∥</sup>*z t*ð Þ∥<sup>2</sup> <sup>≤</sup>2<sup>R</sup> *evt* ð Þ ð Þh i *Bz t*ð Þ*; z t*ð Þ *:*

*<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> <sup>¼</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ � *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ� *S t*ð Þ*z*<sup>0</sup>

Since *B* is locally Lipschitz, there exists a constant positive *L* that depends on

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> ∣≤∣ *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ <sup>∣</sup> <sup>þ</sup> <sup>2</sup>*αL*∥*z t*ðÞ� *S t*ð Þ*z*0∥∥*z*0∥*,* (34)

While from the variation of constants formula and using Schwartz's inequality,

0

Integrating (34) over the interval 0½ � *; T* and taking into account (35) and (36),

<sup>2</sup>*L*<sup>2</sup>

<sup>2</sup>∥*z*0∥ Ð *<sup>T</sup>*

<sup>∥</sup>*U s*ð Þ*z*0∥<sup>2</sup>

*<sup>t</sup>* j j *vUs* ð Þ ð Þ*z*<sup>0</sup>

<sup>þ</sup> *<sup>T</sup>*<sup>1</sup>

Replacing *z*<sup>0</sup> by *U t*ð Þ*z*<sup>0</sup> in (37), and using the superposition properties of the

Now, let us consider the nonlinear semigroup *U t*ð Þ*z*<sup>0</sup> ≔ *z t*ð Þ [1].

<sup>0</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*U t*ð Þ*z*0*; S s*ð Þ*U t*ð Þ*z*<sup>0</sup> <sup>∣</sup>*ds*≤2*αT*<sup>3</sup>

2

*ds* � �<sup>1</sup>

*<sup>t</sup>* j j *vUs* ð Þ ð Þ*z*<sup>0</sup>

<sup>2</sup>∥*U t*ð Þ*z*0<sup>∥</sup> <sup>Ð</sup>*<sup>t</sup>*þ*<sup>T</sup>*

<sup>∥</sup>*z t*ð Þ� *S t*ð Þ*z*0∥ ≤ *L T* <sup>ð</sup>*<sup>T</sup>*

<sup>0</sup> <sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> <sup>∣</sup>*dt* <sup>≤</sup>2*αT*<sup>3</sup>

<sup>þ</sup> *<sup>C</sup>*∗*CBS t*ð Þ*z*<sup>0</sup> � *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; S t*ð Þ*z*<sup>0</sup> *:*

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ ∣≤∣*vzt* ð Þ ð Þ ∣∥*z t*ð Þ∥∥*z*0∥*,* <sup>∀</sup>*t*∈½ � <sup>0</sup>*; <sup>T</sup> :* (35)

j j *vzt* ð Þ ð Þ <sup>2</sup>

*dt* � �<sup>1</sup>

∥*z*0∥ Ð *<sup>T</sup>*

<sup>0</sup> j j *vzt* ð Þ ð Þ <sup>2</sup>

2

<sup>∥</sup>*U s*ð Þ*z*0∥<sup>2</sup>

2

*ds* � �<sup>1</sup>

<sup>∥</sup>*z t*ð Þ∥<sup>2</sup>

<sup>0</sup> j j *vzt* ð Þ ð Þ <sup>2</sup>

<sup>2</sup>*L*<sup>2</sup>

2

<sup>∥</sup>*z t*ð Þ∥<sup>2</sup> *dt* � �<sup>1</sup>

> 2 *:*

<sup>∥</sup>*z t*ð Þ∥<sup>2</sup> *dt* � �<sup>1</sup>

∥*U t*ð Þ*z*0∥

2

*:* (36)

2

(37)

Integrating this inequality, and using hypothesis 2 of Theorem 1.6, it follows that

We have

$$
\langle \chi\_{\;{a}}^{\*} \chi\_{\;{a}} \mathbf{A} z, z \rangle = -\int\_{0}^{1} z'(\boldsymbol{\varkappa}) z(\boldsymbol{\varkappa}) d\boldsymbol{\varkappa} = -\frac{z^{2}(1)}{2} \le 0,
$$

so, the assumption 1 of Theorem 1.5 holds. The operator *B* is compact and verifies

$$\left< \chi\_{\alpha}^{\*} \chi\_{\alpha} B \mathbf{S}(t) z\_{0}, \mathbf{S}(t) z\_{0} \right> = \left( \int\_{0}^{1-t} z\_{0}(\mathbf{x}) d\mathbf{x} \right)^{2}, \quad \mathbf{0} \le t \le 1.$$

Thus

$$\left\langle \chi\_w^\* \chi\_w \text{BS}(t) z\_0, \text{S}(t) z\_0 \right\rangle = \mathbf{0}, \quad \forall t \ge \mathbf{0} \quad \Rightarrow z\_0(\varkappa) = \mathbf{0}, \ \varkappa. \varepsilon \text{ on } \varkappa.$$

Then, the control

$$v(t) = -\frac{\left(\int\_0^1 z(\varkappa, t) d\varkappa\right)^2}{\mathbf{1} + \left(\int\_0^1 z(\varkappa, t) d\varkappa\right)^2},\tag{30}$$

weakly stabilizes the output (29).

#### **3. Stabilization for semilinear systems**

In this section, we give sufficient conditions for exponential, strong and weak stabilization of the output (5). Consider the semilinear system (1) augmented with the output (5).

#### **4. Exponential stabilization**

In this section, we develop sufficient conditions for exponential stabilization of the output (5).

The following result concerns the exponential stabilization of (5).

Theorem 1.6 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* be locally Lipschitz. If the conditions:

$$(\mathbf{1}.\ \%e(\langle \mathbf{C}^\* \mathbf{C} A \mathbf{y}, \mathbf{y} \rangle) \le \mathbf{0}, \ \forall \mathbf{y} \in D(A),$$

2. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CBy; <sup>y</sup>* h i *By; <sup>y</sup>* <sup>≥</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *<sup>H</sup>*,

3. there exist *T, γ* > 0, such that

$$\int\_{0}^{T} |\langle C^\*CBS(t)\eta, S(t)\eta \rangle| dt \ge \gamma \|C\eta\|\_{Y}^2, \quad \forall \eta \in H,\tag{31}$$

hold, then the control

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

$$w(t) = \begin{cases} -\frac{\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle}{\left\| \mathbf{z}(t) \right\|^2}, & \text{if} \quad \mathbf{z}(t) \!\!/ = \mathbf{0}, \\ \mathbf{0}, & \text{if} \quad \mathbf{z}(t) = \mathbf{0}, \end{cases} \tag{32}$$

exponentially stabilizes the output (5).

*w t*ðÞ¼ *χωz t*ð Þ*:* (29)

<sup>2</sup> <sup>≤</sup>0*,*

*,* 0≤*t* ≤1*:*

� �<sup>2</sup> *,* (30)

*Y,* ∀*y* ∈ *H,* (31)

ð Þ *<sup>x</sup> z x*ð Þ*dx* ¼ � *<sup>z</sup>*2ð Þ<sup>1</sup>

*z*0ð Þ *x dx* � �<sup>2</sup>

We have

verifies

Thus

the output (5).

the output (5).

*χ* ∗

*χ* ∗

*χ* ∗

Then, the control

*<sup>ω</sup> χωAz; <sup>z</sup>* � � ¼ �

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*<sup>ω</sup> χωBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> � � <sup>¼</sup>

*v t*ðÞ¼�

*<sup>ω</sup> χωBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup>

weakly stabilizes the output (29).

**4. Exponential stabilization**

locally Lipschitz. If the conditions:

1. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CAy; <sup>y</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *D A*ð Þ,

3. there exist *T, γ* > 0, such that

hold, then the control

**114**

2. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CBy; <sup>y</sup>* h i *By; <sup>y</sup>* <sup>≥</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *<sup>H</sup>*,

ð*T* 0

**3. Stabilization for semilinear systems**

ð1 0 *z*0

so, the assumption 1 of Theorem 1.5 holds. The operator *B* is compact and

ð1�*<sup>t</sup>* 0

� � <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*<sup>t</sup>* <sup>≥</sup><sup>0</sup> ) *<sup>z</sup>*0ð Þ¼ *<sup>x</sup>* <sup>0</sup>*, a:<sup>e</sup>* on *<sup>ω</sup>:*

Ð 1

<sup>1</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

In this section, we give sufficient conditions for exponential, strong and weak stabilization of the output (5). Consider the semilinear system (1) augmented with

In this section, we develop sufficient conditions for exponential stabilization of

Theorem 1.6 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* be

The following result concerns the exponential stabilization of (5).

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*y; S t*ð Þ*<sup>y</sup>* <sup>∣</sup>*dt* <sup>≥</sup>*γ*∥*Cy*∥<sup>2</sup>

<sup>0</sup> *z x*ð Þ *; t dx* � �<sup>2</sup>

<sup>0</sup> *z x*ð Þ *; t dx*

**Proof:** Since ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is a semigroup of contractions, we have

$$\frac{d}{dt} \left\| z(t) \right\|^2 \le 2 \mathcal{R} e(v(t) \langle Bz(t), z(t) \rangle).$$

Integrating this inequality, and using hypothesis 2 of Theorem 1.6, it follows that

$$\|\mathbf{z}(t)\| \le \|\mathbf{z}\_0\|. \tag{33}$$

For all *z*<sup>0</sup> ∈ *H* and *t*≥ 0, we have

$$
\begin{split}
\langle \mathsf{C}^\* \mathsf{C} \mathrm{B} \mathrm{S}(\mathsf{t}) \mathsf{z}\_0, \mathsf{S}(\mathsf{t}) \mathsf{z}\_0 \rangle &= \langle \mathsf{C}^\* \mathsf{C} \mathrm{B} \mathrm{z}(\mathsf{t}), \mathsf{z}(\mathsf{t}) \rangle - \langle \mathsf{C}^\* \mathsf{C} \mathrm{B} \mathrm{z}(\mathsf{t}), \mathsf{z}(\mathsf{t}) - \mathsf{S}(\mathsf{t}) \mathsf{z}\_0 \rangle \\ &+ \langle \mathsf{C}^\* \mathsf{C} \mathrm{B} \mathrm{S}(\mathsf{t}) \mathsf{z}\_0 - \mathsf{C}^\* \mathrm{C} \mathrm{B} \mathrm{z}(\mathsf{t}), \mathsf{S}(\mathsf{t}) \mathsf{z}\_0 \rangle.
\end{split}
$$

Since *B* is locally Lipschitz, there exists a constant positive *L* that depends on ∥*z*0∥ such that

$$|\langle \mathbf{C}^\* \text{CBS}(t) \mathbf{z}\_0, \mathbf{S}(t) \mathbf{z}\_0 \rangle| \le |\langle \mathbf{C}^\* \text{CBS}(t), \mathbf{z}(t) \rangle| + 2aL \|\mathbf{z}(t) - \mathbf{S}(t) \mathbf{z}\_0\| \|\mathbf{z}\_0\|,\tag{34}$$

where *α* is a positive constant.

Using (33), we deduce

$$|\langle C^\*CBz(t), z(t)\rangle| \le |\nu(z(t))| \|z(t)\| \|z\_0\|, \ \forall t \in [0, T]. \tag{35}$$

While from the variation of constants formula and using Schwartz's inequality, we obtain

$$\|\boldsymbol{z}(t) - \boldsymbol{S}(t)\boldsymbol{z}\_{0}\| \le L\left(T\int\_{0}^{T} |\boldsymbol{v}(\boldsymbol{z}(t))|^{2} \|\boldsymbol{z}(t)\|^{2} dt\right)^{\frac{1}{2}}.\tag{36}$$

Integrating (34) over the interval 0½ � *; T* and taking into account (35) and (36), we get

$$\begin{split} \int\_{0}^{T} | \langle C^{\*} \operatorname{CBS}(t) \mathbf{z}\_{0}, \operatorname{S}(t) \mathbf{z}\_{0} \rangle | dt \leq 2\alpha T^{\frac{3}{2}} L^{2} \| \mathbf{z}\_{0} \| \left( \int\_{0}^{T} | \nu(\mathbf{z}(t)) |^{2} \| \mathbf{z}(t) \| ^{2} dt \right)^{\frac{1}{2}} \\ &+ T^{\frac{1}{2}} \| \mathbf{z}\_{0} \| \left( \int\_{0}^{T} | \nu(\mathbf{z}(t)) |^{2} \| \mathbf{z}(t) \| ^{2} dt \right)^{\frac{1}{2}}. \end{split}$$

Now, let us consider the nonlinear semigroup *U t*ð Þ*z*<sup>0</sup> ≔ *z t*ð Þ [1].

Replacing *z*<sup>0</sup> by *U t*ð Þ*z*<sup>0</sup> in (37), and using the superposition properties of the semigroup ð Þ *U t*ð Þ *<sup>t</sup>*≥<sup>0</sup>, we deduce that

$$\int\_{0}^{T} |\langle C^\*CBS(s)U(t)x\_0, S(s)U(t)x\_0 \rangle| ds \le 2aT^{\frac{1}{2}}L^2 \|U(t)x\_0\|$$

$$\times \left(\int\_{t}^{t+T} |\nu(U(s)x\_0)|^2 \|U(s)x\_0\|^2 ds\right)^{\frac{1}{2}}\tag{37}$$

$$+T^{\frac{1}{2}} \|U(t)x\_0\| \left(\int\_{t}^{t+T} |\nu(U(s)x\_0)|^2 \|U(s)x\_0\|^2 ds\right)^{\frac{1}{2}}$$

Thus, by using (31) and (37), it follows that

$$\|\chi\|\mathrm{CU}(t)\mathrm{z}\_{0}\|\_{Y} \le \mathrm{M}\left(\int\_{t}^{t+T} |\nu(U(s)\mathrm{z}\_{0})|^{2} \|\boldsymbol{U}(s)\mathrm{z}\_{0}\|^{2} ds\right)^{\frac{1}{2}},\tag{38}$$

hold, then the control

strongly stabilizes the output (5).

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

Integrating this inequality, gives

Thus

deduce

ð*T* 0

**117**

ð*T* 0

> <sup>þ</sup> *<sup>T</sup>*<sup>1</sup> 2 ð*<sup>t</sup>*þ*<sup>T</sup> t*

By (43), we get

completes the proof.

**Proof:** From hypothesis 1 of Theorem 1.7, we obtain

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*dt* <sup>∥</sup>*Cz t*ð Þ∥<sup>2</sup>

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup>

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup>

2 ð*T* 0

From the variation of constants formula and using Schwartz's inequality, we

Integrating (34) over the interval 0½ � *; T* and taking into account (44), we obtain

*T*3 <sup>2</sup>∥*z*0∥<sup>2</sup>

> 2 *:*

Replacing *z*<sup>0</sup> by *z t*ð Þ and using the superposition property of the solution, we get

2 *:*

*T*3 <sup>2</sup>∥*z*0∥<sup>2</sup>

From (40) and (46), we deduce that ∥*Cz t*ð Þ∥*<sup>Y</sup>* ! 0, as *t* ! þ∞, which

locally Lipschitz and the assumptions 1, 2 and 3 of Theorem 1.7 hold, then the

control (41) strongly stabilizes the output (5) with decay estimate

Proposition 1.8 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* be

*d*

2 ð*t* 0

> ðþ<sup>∞</sup> 0

<sup>∥</sup>*z t*ðÞ� *S t*ð Þ*z*0∥ ≤*LT*<sup>1</sup>

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z*0*; S s*ð Þ*z*<sup>0</sup> <sup>∣</sup>*ds*<sup>≤</sup> <sup>2</sup>*αL*<sup>2</sup>

<sup>0</sup> 〈*C*<sup>∗</sup> j j *CBz s*ð Þ*; z s*ð Þ〉 <sup>2</sup> *ds* � �<sup>1</sup>

〈*C*<sup>∗</sup> j j *CBz s*ð Þ*; z s*ð Þ〉 <sup>2</sup> *ds* � �<sup>1</sup>

<sup>þ</sup> *<sup>T</sup>*<sup>1</sup> 2 Ð *T*

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z t*ð Þ*; S s*ð Þ*z t*ð Þ <sup>∣</sup>*ds* <sup>≤</sup>2*αL*<sup>2</sup>

ð*<sup>t</sup>*þ*<sup>T</sup> t*

*v t*ðÞ¼� *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ *,* (41)

*ds*≤ ∥*Cz*ð Þ <sup>0</sup> <sup>∥</sup><sup>2</sup>

*<sup>C</sup>*<sup>∗</sup> j j h i *CBz s*ð Þ*; z s*ð Þ <sup>2</sup> *ds* � �<sup>1</sup>

> ð*T* 0

ð*<sup>t</sup>*þ*<sup>T</sup> t*

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*z t*ð Þ*; S s*ð Þ*z t*ð Þ <sup>∣</sup>*ds* ! <sup>0</sup>*,* as *<sup>t</sup>* ! þ∞*:* (46)

*Y:*

*ds*< þ ∞*,* (43)

2

〈*C*<sup>∗</sup> j j *CBz s*ð Þ*; z s*ð Þ〉 <sup>2</sup> *ds* � �<sup>1</sup>

〈*C*<sup>∗</sup> j j *CBz s*ð Þ*; z s*ð Þ〉 <sup>2</sup> *ds* � �<sup>1</sup>

*:* (44)

2

2

(45)

*:* (42)

*<sup>Y</sup>* <sup>≤</sup> � <sup>2</sup> *<sup>C</sup>*<sup>∗</sup> j j h i *CBz t*ð Þ*; z t*ð Þ <sup>2</sup>

where *<sup>M</sup>* <sup>¼</sup> <sup>2</sup>*αTL*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �*T*<sup>1</sup> <sup>2</sup> is a non-negative constant depending on ∥*z*0∥ and *T*. From hypothesis 1 of Theorem 1.6, we have

$$\frac{d}{dt} \|CU(t)\mathbf{z}\_0\|\_Y^2 \le -2|v(U(t)\mathbf{z}\_0)|^2 \|U(t)\mathbf{z}\_0\|^2. \tag{39}$$

Integrating (39) from *nT* and ð Þ *n* þ 1 *T, n*ð Þ ∈ N , we obtain

$$\left\| \|CU(nT)\mathbf{z}\_0\|\right\|\_Y^2 - \left\|CU((n+1)T)\mathbf{z}\_0\right\|\_Y^2 \ge 2\int\_{nT}^{(n+1)T} |v(U(s)\mathbf{z}\_0)|^2 \|U(s)\mathbf{z}\_0\|^2 ds.$$

Using (38), (39) and the fact that ∥*CU t*ð Þ*z*0∥*<sup>Y</sup>* decreases, it follows

$$\|\left(1+2\left(\frac{\gamma}{M}\right)^2\right)\|CU((n+1)T)z\_0\|\_Y^2 \le \|CU(nT)z\_0\|\_Y^2.$$

Then

$$\|\mathbb{C}U((n+1)T)\mathbb{Z}\_0\|\_Y \le \beta \|\mathbb{C}U(nT)\mathbb{Z}\_0\|\_Y,$$

where *<sup>β</sup>* <sup>¼</sup> <sup>1</sup> <sup>1</sup>þ<sup>2</sup> *<sup>γ</sup>* ð Þ *<sup>M</sup>* <sup>2</sup> � �<sup>1</sup> 2 *:* By recurrence, we show that <sup>∥</sup>*CU nT* ð Þ*z*0∥*<sup>Y</sup>* <sup>≤</sup>*β<sup>n</sup>*∥*Cz*0∥*Y*. Taking *<sup>n</sup>* <sup>¼</sup> *<sup>E</sup> <sup>t</sup> T* � � the integer part of *<sup>t</sup> <sup>T</sup>*, we deduce that

$$\|CU(t)z\_0\|\_Y \le \mathbf{Re}^{-\sigma t} \|z\_0\|\_{\ast}$$

where *<sup>R</sup>* <sup>¼</sup> *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> <sup>2</sup> *<sup>γ</sup> M* � �<sup>2</sup> � �<sup>1</sup> 2 , with *<sup>α</sup>*<sup>&</sup>gt; 0 and *<sup>σ</sup>* <sup>¼</sup> ln 1þ<sup>2</sup> *<sup>γ</sup>* ð Þ*<sup>M</sup>* <sup>2</sup> � � <sup>2</sup>*<sup>T</sup>* >0, which achieves the proof.

#### **4.1 Strong stabilization**

The following result provides sufficient conditions for strong stabilization of the output (5).

Theorem 1.7 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* be locally Lipschitz. If the conditions:


$$\int\_{0}^{T} |\langle \mathbf{C}^\* \mathbf{C} \mathbf{S} \mathbf{S}(t) \mathbf{y}, \mathbf{S}(t) \mathbf{y} \rangle| dt \ge \gamma \| \mathbf{C} \mathbf{y} \|\_{Y}^2, \quad \forall \mathbf{y} \in H,\tag{40}$$

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

hold, then the control

$$v(t) = -\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} z(t), z(t) \rangle,\tag{41}$$

strongly stabilizes the output (5). **Proof:** From hypothesis 1 of Theorem 1.7, we obtain

$$\frac{d}{dt} \|\mathbf{C}z(t)\|\_{Y}^{2} \le -2 \left| \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B}z(t), z(t) \rangle \right|^{2}. \tag{42}$$

Integrating this inequality, gives

$$2\int\_0^t \left| \langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(s), \mathbf{z}(s) \rangle \right|^2 ds \le \| \mathbf{C} \mathbf{z}(\mathbf{0}) \|\_Y^2.$$

Thus

Thus, by using (31) and (37), it follows that

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*γ*∥*CU t*ð Þ*z*0∥*<sup>Y</sup>* ≤ *M*

From hypothesis 1 of Theorem 1.6, we have

*dt* <sup>∥</sup>*CU t*ð Þ*z*0∥<sup>2</sup>

Integrating (39) from *nT* and ð Þ *n* þ 1 *T, n*ð Þ ∈ N , we obtain

Using (38), (39) and the fact that ∥*CU t*ð Þ*z*0∥*<sup>Y</sup>* decreases, it follows

<sup>∥</sup>*CU n* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup> <sup>z</sup>*0∥<sup>2</sup>

∥*CU n* ð Þ ð Þ þ 1 *T z*0∥*<sup>Y</sup>* ≤ *β*∥*CU nT* ð Þ*z*0∥*Y,*

<sup>∥</sup>*CU t*ð Þ*z*0∥*<sup>Y</sup>* <sup>≤</sup>Re�*σ<sup>t</sup>*

, with *<sup>α</sup>*<sup>&</sup>gt; 0 and *<sup>σ</sup>* <sup>¼</sup> ln 1þ<sup>2</sup> *<sup>γ</sup>* ð Þ*<sup>M</sup>*

The following result provides sufficient conditions for strong stabilization of the

Theorem 1.7 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H* and *B* be

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*y; S t*ð Þ*<sup>y</sup>* <sup>∣</sup>*dt* <sup>≥</sup>*γ*∥*Cy*∥<sup>2</sup>

*<sup>T</sup>*, we deduce that

∥*z*0∥*,*

<sup>2</sup> � �

*<sup>Y</sup>* � <sup>∥</sup>*CU n* ð Þ ð Þ <sup>þ</sup> <sup>1</sup> *<sup>T</sup> <sup>z</sup>*0∥<sup>2</sup>

By recurrence, we show that <sup>∥</sup>*CU nT* ð Þ*z*0∥*<sup>Y</sup>* <sup>≤</sup>*β<sup>n</sup>*∥*Cz*0∥*Y*.

� � the integer part of *<sup>t</sup>*

2

*M* � �<sup>2</sup> � �<sup>1</sup>

*d*

<sup>1</sup> <sup>þ</sup> <sup>2</sup> *<sup>γ</sup> M* � �<sup>2</sup> � �

where *<sup>M</sup>* <sup>¼</sup> <sup>2</sup>*αTL*<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �*T*<sup>1</sup>

<sup>∥</sup>*CU nT* ð Þ*z*0∥<sup>2</sup>

Then

proof.

output (5).

**116**

where *<sup>β</sup>* <sup>¼</sup> <sup>1</sup>

Taking *<sup>n</sup>* <sup>¼</sup> *<sup>E</sup> <sup>t</sup>*

where *<sup>R</sup>* <sup>¼</sup> *<sup>α</sup>* <sup>1</sup> <sup>þ</sup> <sup>2</sup> *<sup>γ</sup>*

**4.1 Strong stabilization**

locally Lipschitz. If the conditions:

1. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CAy; <sup>y</sup>* <sup>≤</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *D A*ð Þ,

3. there exist *T, γ* > 0, such that

2. <sup>R</sup> *e C*<sup>∗</sup> ð Þ h i *CBy; <sup>y</sup>* h i *By; <sup>y</sup>* <sup>≥</sup>0*,* <sup>∀</sup>*y*<sup>∈</sup> *<sup>H</sup>*,

ð*T* 0

<sup>1</sup>þ<sup>2</sup> *<sup>γ</sup>* ð Þ *<sup>M</sup>* <sup>2</sup> � �<sup>1</sup> 2 *:*

*T*

ð*t*þ*<sup>T</sup> t*

j j *vUs* ð Þ ð Þ*z*<sup>0</sup>

*<sup>Y</sup>* ≤ � 2j j *vUt* ð Þ ð Þ*z*<sup>0</sup>

*<sup>Y</sup>* ≥2

2

2

ðð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>T</sup> nT*

� �<sup>1</sup>

<sup>∥</sup>*U s*ð Þ*z*0∥<sup>2</sup>

<sup>2</sup> is a non-negative constant depending on ∥*z*0∥ and *T*.

<sup>∥</sup>*U t*ð Þ*z*0∥<sup>2</sup>

j j *vUs* ð Þ ð Þ*z*<sup>0</sup>

*<sup>Y</sup>* ≤ ∥*CU nT* ð Þ*z*0∥<sup>2</sup>

2

*Y:*

<sup>2</sup>*<sup>T</sup>* >0, which achieves the

*Y,* ∀*y* ∈ *H,* (40)

*ds*

2

*,* (38)

*:* (39)

<sup>∥</sup>*U s*ð Þ*z*0∥<sup>2</sup>

*ds:*

$$\int\_0^{+\infty} \left| \langle C^\* CBz(s), z(s) \rangle \right|^2 ds < +\infty,\tag{43}$$

From the variation of constants formula and using Schwartz's inequality, we deduce

$$\|\|x(t) - S(t)x\_0\|\| \le LT^{\frac{1}{2}} \left( \int\_0^T |\langle C^\*CBx(s), x(s)\rangle|^2 ds \right)^{\frac{1}{2}}.\tag{44}$$

Integrating (34) over the interval 0½ � *; T* and taking into account (44), we obtain

$$\begin{split} \int\_{0}^{T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{S}(s) \mathbf{z}\_{0}, \mathbf{S}(s) \mathbf{z}\_{0} \rangle | ds &\leq 2aL^{2}T^{\frac{3}{2}} \| \mathbf{z}\_{0} \| ^{2} \left( \int\_{0}^{T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{z}(s), \mathbf{z}(s) \rangle |^{2} ds \right)^{\frac{1}{2}} \\ &+ T^{\frac{1}{2}} \left( \int\_{0}^{T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{z}(s), \mathbf{z}(s) \rangle |^{2} ds \right)^{\frac{1}{2}}. \end{split}$$

Replacing *z*<sup>0</sup> by *z t*ð Þ and using the superposition property of the solution, we get

$$\begin{split} \int\_{0}^{T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{S}(\mathbf{s}) \mathbf{z}(t), \mathbf{S}(\mathbf{s}) \mathbf{z}(t) \rangle | d\mathbf{s} &\leq 2a\mathbf{L}^{2}T^{\frac{3}{2}} \| \mathbf{z}\_{0} \|^{2} \left( \int\_{t}^{t+T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{z}(\mathbf{s}), \mathbf{z}(\mathbf{s}) \rangle |^{2} d\mathbf{s} \right)^{\frac{1}{2}} \\ &+ T^{\frac{1}{2}} \left( \int\_{t}^{t+T} | \langle \mathbf{C}^{\*} \, \mathbf{C} \mathbf{B} \mathbf{z}(\mathbf{s}), \mathbf{z}(\mathbf{s}) \rangle |^{2} d\mathbf{s} \right)^{\frac{1}{2}}. \end{split} \tag{45}$$

By (43), we get

$$\int\_{t}^{t+T} | \langle C^\*CBS(s)z(t), S(s)z(t) \rangle | ds \to \mathbf{0}, \text{as } t \to +\infty. \tag{46}$$

From (40) and (46), we deduce that ∥*Cz t*ð Þ∥*<sup>Y</sup>* ! 0, as *t* ! þ∞, which completes the proof.

Proposition 1.8 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* be locally Lipschitz and the assumptions 1, 2 and 3 of Theorem 1.7 hold, then the control (41) strongly stabilizes the output (5) with decay estimate

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\|\mathbb{C}\mathbf{z}(t)\|\_{Y} = O\left(t^{-\frac{4}{2}}\right), \text{ as } t \to +\infty. \tag{47}$$

**4.2 Weak stabilization**

hold, then the control

*tn* ! þ∞, as *n* ! þ∞.

∀*t*≥0, we get

quence still denoted by *z tγ*ð Þ *<sup>n</sup>*

From (46), we have

ð*T* 0

*t* ! þ∞, which achieves the proof.

8 < :

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

control operator and *<sup>v</sup>*ð Þ*:* <sup>∈</sup>*L*<sup>2</sup>

*D A*ð Þ¼ *<sup>z</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup>

contractions

**119**

*<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

ð Þ <sup>Ω</sup> , *Az* ¼ � *<sup>∂</sup><sup>z</sup>*

*<sup>∂</sup><sup>t</sup>* ¼ � *<sup>∂</sup>z x*ð Þ *; <sup>t</sup>*

ð Þ *S t*ð Þ*z*<sup>0</sup> ð Þ¼ *x*

weakly stabilizes the output (5).

From (33), the subsequence *z tγ*ð Þ *<sup>n</sup>*

lim*<sup>n</sup>*!þ<sup>∞</sup> *<sup>C</sup>*∗*CBS t*ð Þ*z tγ*ð Þ *<sup>n</sup>*

*<sup>C</sup>*∗*CBS s*ð Þ*z tγ*ð Þ *<sup>n</sup>*

Theorem 1.7 and

The following result discusses the weak stabilization of the output (5). Theorem 1.9 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* be

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

locally Lipschitz and weakly sequentially continuous. If assumptions 1, 2 of

*<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*y; S t*ð Þ*<sup>y</sup>* <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*t*≥<sup>0</sup> ) *Cy* <sup>¼</sup> <sup>0</sup>*,* (51)

**Proof:** Let us consider *ψ* ∈*Y* and ð Þ *tn* ≥ 0 be a sequence of real numbers such that

Since *C* is continuous, *B* is weakly sequentially continuous and *S t*ð Þ is continuous

� � � � <sup>¼</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*φ; S t*ð Þ*<sup>φ</sup> :*

� � � � *ds* ! <sup>0</sup>*,* as *<sup>n</sup>* ! þ∞*:*

*<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*φ; S t*ð Þ*<sup>φ</sup>* <sup>¼</sup> <sup>0</sup>*,* for all *<sup>t</sup>*<sup>≥</sup> <sup>0</sup>*,*

which implies, according to (51), that *Cφ* ¼ 0, and hence *hn* ! 0, as *t* ! þ∞. We deduce that h i *Cz t*ð Þ*; ψ <sup>Y</sup>* ! 0, as *t* ! þ∞. In other words *Cz t*ð Þ ⇀ 0, as

þ *v t*ð Þ*Bz x*ð Þ *; t , x*∈ Ω*, t*>0*,*

ð Þ 0*;* þ∞ . The operator *A* generates a semigroup of

*z*0ð Þ *x* � *t ,* if *x*≥*t,* 0*,* if *x*<*t:* (53)

Ð 1

<sup>0</sup> *z x*ð Þ*dx*∣ the

Using (42), we deduce that the sequence *hn* ¼ h i *Cz t*ð Þ*<sup>n</sup> ; ψ <sup>Y</sup>* is bounded.

� � such that *z tγ*ð Þ *<sup>n</sup>*

� �*; S t*ð Þ*z tγ*ð Þ *<sup>n</sup>*

� �*; S s*ð Þ*z tγ*ð Þ *<sup>n</sup>*

Using the dominated convergence Theorem, we deduce that

Example 4 Let us consider the system defined in Ω ¼�0*,* þ ∞½ by

*z x*ð Þ¼ *;* 0 *z*0ð Þ *x , x*∈ Ω*,*

*<sup>∂</sup><sup>x</sup>* with domain

�

Let *ω* ¼�0*,* 1½ be a subregion of Ω and system (53) is augmented with the output

ð Þj <sup>Ω</sup> *<sup>z</sup>*ð Þ¼ <sup>0</sup> <sup>0</sup>*; z x*ð Þ! <sup>0</sup>*;* as *<sup>x</sup>* ! þ<sup>∞</sup> � �, *Bz* <sup>¼</sup> <sup>∣</sup>

*∂x*

Let *hγ*ð Þ *<sup>n</sup>* be an arbitrary convergent subsequence of *hn*.

*v t*ðÞ¼� *<sup>C</sup>*<sup>∗</sup> h i *CBz t*ð Þ*; z t*ð Þ *,* (52)

� � is bounded in *H*, so we can extract a subse-

� � ⇀ *<sup>φ</sup>*<sup>∈</sup> *<sup>H</sup>*, as *<sup>n</sup>* ! þ∞.

**Proof:** Using (45), we get

$$\int\_{0}^{T} | \langle C^\*CBS(s)U(t)z\_0, S(s)U(t)z\_0 \rangle | ds \le \theta \sqrt{\xi(t)},\tag{48}$$

where *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*αTL*<sup>2</sup> <sup>∥</sup>*z*0∥<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �*T*<sup>1</sup> <sup>2</sup> and *ξ*ðÞ¼ *t* ð*t*þ*<sup>T</sup> t <sup>C</sup>*<sup>∗</sup> j j h i *CBU s*ð Þ*z*0*; U s*ð Þ*z*<sup>0</sup> 2 *ds* � �. From (40) and (48), we deduce that

$$\varrho\sqrt{\xi(nT)} \ge \|CU(nT)x\_0\|\_Y^2, \quad \forall n \ge 0,\tag{49}$$

where <sup>ϱ</sup> <sup>¼</sup> <sup>1</sup> *γ θ:*

Integrating the above inequality gives

$$\frac{d}{dt} \|CU(t)\mathbf{z}\_0\|\_Y^2 \le -2 \left| \langle \mathbf{C}^\* \mathbf{C}BU(t)\mathbf{z}\_0, U(t)\mathbf{z}\_0 \rangle \right|^2,$$

from *nT* to ð Þ *n* þ 1 *T*, ð Þ *n* ∈ N and using (49), we obtain

$$\|\|CU(nT)z\_0\|\|\_Y^2 - \|CU(nT+T)z\_0\|\|\_Y^2 \ge 2\xi(nT), \quad \forall n \ge 0.$$

We obtain

$$\|\varrho^2\|\mathrm{CU}(nT+T)\mathrm{z}\_0\|\_Y^2 - \varrho^2\|\mathrm{CU}(nT)\mathrm{z}\_0\|\_Y^2 \le -2\|\mathrm{CU}(nT)\mathrm{z}\_0\|\_Y^4, \quad \forall n \ge 0. \tag{50}$$

Let us introduce the sequence *rn* <sup>¼</sup> <sup>∥</sup>*CU nT* ð Þ*z*0∥<sup>2</sup> *Y,* ∀*n*≥ 0*:* Using (50), we deduce that

$$\frac{r\_n - r\_{n+1}}{r\_n^2} \ge \frac{2}{\mathfrak{q}^2}, \quad \forall n \ge 0.$$

Since the sequence ð Þ *rn* decreases, we get

$$\frac{r\_n - r\_{n+1}}{r\_n.r\_{n+1}} \ge \frac{2}{\mathfrak{q}^2}, \quad \forall n \ge 0,$$

and also

$$\frac{1}{r\_{n+1}} - \frac{1}{r\_n} \ge \frac{2}{\mathfrak{q}^2}, \quad \forall n \ge 0.$$

We deduce that

$$r\_n \le \frac{r\_0}{\frac{2r\_0}{q^\ell}n + 1}, \quad \forall n \ge 0.$$

Finally, introducing the integer part *<sup>n</sup>* <sup>¼</sup> *<sup>E</sup> <sup>t</sup> T* � � and from (42), the function *t* ! ∥*CU t*ð Þ*z*0∥*<sup>Y</sup>* decreases. We deduce the estimate

$$\|\mathbb{C}\mathbf{z}(t)\|\_{Y} = O\left(t^{-1/2}\right), \quad \text{as } t \to +\infty.$$

#### **4.2 Weak stabilization**

<sup>∥</sup>*Cz t*ð Þ∥*<sup>Y</sup>* <sup>¼</sup> *O t*�<sup>1</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

**Proof:** Using (45), we get

where *<sup>θ</sup>* <sup>¼</sup> <sup>2</sup>*αTL*<sup>2</sup>

where <sup>ϱ</sup> <sup>¼</sup> <sup>1</sup>

We obtain

and also

**118**

We deduce that

<sup>∥</sup>*CU nT* ð Þ <sup>þ</sup> *<sup>T</sup> <sup>z</sup>*0∥<sup>2</sup>

Using (50), we deduce that

ϱ2

*γ θ:*

ð*T* 0

<sup>∥</sup>*z*0∥<sup>2</sup> <sup>þ</sup> <sup>1</sup> � �*T*<sup>1</sup>

ϱ ffiffiffiffiffiffiffiffiffiffiffiffi

From (40) and (48), we deduce that

Integrating the above inequality gives

*dt* <sup>∥</sup>*CU t*ð Þ*z*0∥<sup>2</sup>

*<sup>Y</sup>* � <sup>ϱ</sup><sup>2</sup>

Since the sequence ð Þ *rn* decreases, we get

Let us introduce the sequence *rn* <sup>¼</sup> <sup>∥</sup>*CU nT* ð Þ*z*0∥<sup>2</sup>

from *nT* to ð Þ *n* þ 1 *T*, ð Þ *n* ∈ N and using (49), we obtain

*d*

<sup>∥</sup>*CU nT* ð Þ*z*0∥<sup>2</sup>

2 � �

<sup>∣</sup> *<sup>C</sup>*<sup>∗</sup> h i *CBS s*ð Þ*U t*ð Þ*z*0*; S s*ð Þ*U t*ð Þ*z*<sup>0</sup> <sup>∣</sup>*ds* <sup>≤</sup>*<sup>θ</sup>* ffiffiffiffiffiffiffiffi

ð*t*þ*<sup>T</sup> t*

*<sup>Y</sup>* <sup>≤</sup> � <sup>2</sup> *<sup>C</sup>*<sup>∗</sup> j j h i *CBU t*ð Þ*z*0*; U t*ð Þ*z*<sup>0</sup>

<sup>2</sup> and *ξ*ðÞ¼ *t*

*<sup>ξ</sup>*ð Þ *nT* <sup>p</sup> ≥ ∥*CU nT* ð Þ*z*0∥<sup>2</sup>

*<sup>Y</sup>* � <sup>∥</sup>*CU nT* ð Þ <sup>þ</sup> *<sup>T</sup> <sup>z</sup>*0∥<sup>2</sup>

<sup>∥</sup>*CU nT* ð Þ*z*0∥<sup>2</sup>

*rn* � *rn*þ<sup>1</sup> *r*2 *n*

*rn* � *rn*þ<sup>1</sup> *rn:rn*þ<sup>1</sup>

1 *rn*þ<sup>1</sup>

*rn* ≤

<sup>∥</sup>*Cz t*ð Þ∥*<sup>Y</sup>* <sup>¼</sup> *O t*�1*=*<sup>2</sup> � �

Finally, introducing the integer part *<sup>n</sup>* <sup>¼</sup> *<sup>E</sup> <sup>t</sup>*

*t* ! ∥*CU t*ð Þ*z*0∥*<sup>Y</sup>* decreases. We deduce the estimate

� 1 *rn* ≥ 2

> *r*0 2*r*<sup>0</sup> <sup>ϱ</sup><sup>2</sup> *n* þ 1

≥ 2

≥ 2 *, as t* ! þ∞*:* (47)

*<sup>C</sup>*<sup>∗</sup> j j h i *CBU s*ð Þ*z*0*; U s*ð Þ*z*<sup>0</sup>

*Y,* ∀*n* ≥0*,* (49)

2 *,*

*Y,* ∀*n*≥0*:* (50)

*<sup>Y</sup>* ≥2*ξ*ð Þ *nT ,* ∀*n*≥ 0*:*

*Y,* ∀*n*≥ 0*:*

*<sup>Y</sup>* <sup>≤</sup> � <sup>2</sup>∥*CU nT* ð Þ*z*0∥<sup>4</sup>

<sup>ϱ</sup><sup>2</sup> *,* <sup>∀</sup>*<sup>n</sup>* <sup>≥</sup>0*:*

<sup>ϱ</sup><sup>2</sup> *,* <sup>∀</sup>*n*<sup>≥</sup> <sup>0</sup>*,*

<sup>ϱ</sup><sup>2</sup> *,* <sup>∀</sup>*<sup>n</sup>* <sup>≥</sup>0*:*

*,* ∀*n*≥0*:*

*T*

*, as t* ! þ∞*:*

� � and from (42), the function

� �

*<sup>ξ</sup>*ð Þ*<sup>t</sup>* <sup>p</sup> *,* (48)

2 *ds*

.

The following result discusses the weak stabilization of the output (5).

Theorem 1.9 Let *A* generate a semigroup ð Þ *S t*ð Þ *<sup>t</sup>*≥<sup>0</sup> of contractions on *H*, *B* be locally Lipschitz and weakly sequentially continuous. If assumptions 1, 2 of Theorem 1.7 and

$$<\langle C^\*CBS(t)y, S(t)y \rangle = 0, \quad \forall t \ge 0 \Rightarrow Cy = 0,\tag{51}$$

hold, then the control

$$v(t) = -\langle \mathbf{C}^\* \mathbf{C} \mathbf{B} \mathbf{z}(t), \mathbf{z}(t) \rangle,\tag{52}$$

weakly stabilizes the output (5).

**Proof:** Let us consider *ψ* ∈*Y* and ð Þ *tn* ≥ 0 be a sequence of real numbers such that *tn* ! þ∞, as *n* ! þ∞.

Using (42), we deduce that the sequence *hn* ¼ h i *Cz t*ð Þ*<sup>n</sup> ; ψ <sup>Y</sup>* is bounded.

Let *hγ*ð Þ *<sup>n</sup>* be an arbitrary convergent subsequence of *hn*.

From (33), the subsequence *z tγ*ð Þ *<sup>n</sup>* � � is bounded in *H*, so we can extract a subsequence still denoted by *z tγ*ð Þ *<sup>n</sup>* � � such that *z tγ*ð Þ *<sup>n</sup>* � � ⇀ *<sup>φ</sup>*<sup>∈</sup> *<sup>H</sup>*, as *<sup>n</sup>* ! þ∞.

Since *C* is continuous, *B* is weakly sequentially continuous and *S t*ð Þ is continuous ∀*t*≥0, we get

$$\lim\_{n \to +\infty} \left< \mathbf{C}^\* \mathbf{C} \mathbf{S} \mathbf{S}(t) \mathbf{z} \left( \mathbf{t}\_{\gamma(n)} \right), \mathbf{S}(t) \mathbf{z} \left( \mathbf{t}\_{\gamma(n)} \right) \right> = \left< \mathbf{C}^\* \mathbf{C} \mathbf{S} \mathbf{S}(t) \boldsymbol{\rho}, \mathbf{S}(t) \boldsymbol{\rho} \right>.$$

From (46), we have

$$\int\_0^T \langle \mathcal{C}^\* \mathcal{C} \mathcal{B} \mathcal{S}(s) z \left( t\_{\gamma(n)} \right), \mathcal{S}(s) z \left( t\_{\gamma(n)} \right) \rangle ds \to \mathbf{0}, \quad \text{as} \ n \to +\infty.$$

Using the dominated convergence Theorem, we deduce that

*<sup>C</sup>*<sup>∗</sup> h i *CBS t*ð Þ*φ; S t*ð Þ*<sup>φ</sup>* <sup>¼</sup> <sup>0</sup>*,* for all *<sup>t</sup>*<sup>≥</sup> <sup>0</sup>*,*

which implies, according to (51), that *Cφ* ¼ 0, and hence *hn* ! 0, as *t* ! þ∞. We deduce that h i *Cz t*ð Þ*; ψ <sup>Y</sup>* ! 0, as *t* ! þ∞. In other words *Cz t*ð Þ ⇀ 0, as

*t* ! þ∞, which achieves the proof.

Example 4 Let us consider the system defined in Ω ¼�0*,* þ ∞½ by

$$\begin{cases} \frac{\partial \mathbf{z}(\mathbf{x},t)}{\partial t} = -\frac{\partial \mathbf{z}(\mathbf{x},t)}{\partial \mathbf{x}} + \nu(t) \mathbf{B} \mathbf{z}(\mathbf{x},t), & \mathbf{x} \in \Omega, \ t > 0, \\\mathbf{z}(\mathbf{x},0) = \mathbf{z}\_0(\mathbf{x}), & \mathbf{x} \in \Omega, \end{cases} \tag{53}$$

where *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ <sup>Ω</sup> , *Az* ¼ � *<sup>∂</sup><sup>z</sup> <sup>∂</sup><sup>x</sup>* with domain

*D A*ð Þ¼ *<sup>z</sup>*<sup>∈</sup> *<sup>H</sup>*<sup>1</sup> ð Þj <sup>Ω</sup> *<sup>z</sup>*ð Þ¼ <sup>0</sup> <sup>0</sup>*; z x*ð Þ! <sup>0</sup>*;* as *<sup>x</sup>* ! þ<sup>∞</sup> � �, *Bz* <sup>¼</sup> <sup>∣</sup> Ð 1 <sup>0</sup> *z x*ð Þ*dx*∣ the control operator and *<sup>v</sup>*ð Þ*:* <sup>∈</sup>*L*<sup>2</sup> ð Þ 0*;* þ∞ . The operator *A* generates a semigroup of contractions

$$(\mathcal{S}(t)z\_0)(\mathbf{x}) = \begin{cases} z\_0(\mathbf{x} - t), & \text{if} \quad \mathbf{x} \ge t, \\ \mathbf{0}, & \text{if} \quad \mathbf{x} < t. \end{cases}$$

Let *ω* ¼�0*,* 1½ be a subregion of Ω and system (53) is augmented with the output

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$
\omega(t) = \chi\_o \mathbf{z}(t). \tag{54}
$$

The operator *B* is sequentially continuous and verifies

$$\left\langle \chi\_w^\* \chi\_w B \mathbf{S}(\mathbf{t}) \mathbf{z}\_0, \mathbf{S}(\mathbf{t}) \mathbf{z}\_0 \right\rangle = \left| \int\_0^{1-t} z\_0(\mathbf{x}) d\mathbf{x} \right| \int\_0^{1-t} z\_0(\mathbf{x}) d\mathbf{x}, \quad \mathbf{0} \le t \le 1.$$

Thus

$$\left\langle \chi\_{\;\;\alpha}^{\*} \chi\_{\;\alpha} \text{BS}(t) \mathbf{z}\_{0}, \mathbf{S}(t) \mathbf{z}\_{0} \right\rangle = \mathbf{0}, \quad \forall t \ge \mathbf{0} \quad \Rightarrow \mathbf{z}\_{0}(\mathbf{x}) = \mathbf{0} \quad \mathbf{a}. \mathbf{e} \ \mathbf{x} \in ] \mathbf{0}, \mathbf{1}[, \quad \text{i.e.} \quad \chi] \mathbf{0}, \mathbf{1}[\mathbf{z}\_{0} = \mathbf{0}. \mathbf{0}]$$

Then, the control

$$w(t) = -|\int\_0^1 z(\varkappa, t)d\varkappa|\int\_0^1 z(\varkappa, t)d\varkappa,\tag{55}$$

weakly stabilizes the output (54).

#### **5. Simulations**

Consider system (53) with *z x*ð Þ¼ *;* 0 sin ð Þ *πx* , and augmented with the output (54).

For *ω* ¼�0*,* 2½, we have

**Figure 1** shows that the output (54) is weakly stabilized on *ω* with error equals <sup>6</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>4</sup> and the evolution of control is given by **Figure 2**.

For *ω* ¼�0*,* 3½, we have

**Figure 3** shows that the output (54) is weakly stabilized on *ω* with error equals <sup>9</sup>*:*<sup>88</sup> � <sup>10</sup>�<sup>4</sup> and the evolution of control is given by **Figure 4**.

**Remark 2.** It is clear that the control (55) stabilizes the state on *ω*, but do not

In this work, we discuss the question of output stabilization for a class of semilinear systems. Under sufficient conditions, we obtain controls depending on

systems. This work gives an opening to others questions; this is the case of output stabilization for hyperbolic semilinear systems. This will be the purpose of a future

the output operator that strongly and weakly stabilizes the output of such

take into account the residual part Ω *ω*.

**6. Conclusions**

*The stabilization on ω* ¼�0*,* 3½*.*

**Figure 3.**

**Figure 2.**

*The evolution control in the interval* �0*,* 8�*.*

*On the Stabilization of Infinite Dimensional Semilinear Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

research paper.

**121**

**Figure 1.** *The stabilization on ω* ¼�0*,* 2½*.*

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

**Figure 2.** *The evolution control in the interval* �0*,* 8�*.*

*w t*ðÞ¼ *χωz t*ð Þ*:* (54)

*z*0ð Þ *x dx,* 0≤*t*≤1*:*

*z x*ð Þ *; t dx,* (55)

The operator *B* is sequentially continuous and verifies

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*v t*ðÞ¼�∣

<sup>6</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>4</sup> and the evolution of control is given by **Figure 2**.

<sup>9</sup>*:*<sup>88</sup> � <sup>10</sup>�<sup>4</sup> and the evolution of control is given by **Figure 4**.

ð1 0

ð1�*<sup>t</sup>* 0

*z*0ð Þ *x dx*∣

� � <sup>¼</sup> <sup>0</sup>*,* <sup>∀</sup>*t*<sup>≥</sup> <sup>0</sup> ) *<sup>z</sup>*0ð Þ¼ *<sup>x</sup>* 0 a*:*<sup>e</sup> *<sup>x</sup>*∈�0*,* <sup>1</sup>½*,* <sup>i</sup>*:*<sup>e</sup> *<sup>χ</sup>*�0*,* <sup>1</sup>½*z*<sup>0</sup> <sup>¼</sup> <sup>0</sup>*:*

*z x*ð Þ *; t dx*∣

Consider system (53) with *z x*ð Þ¼ *;* 0 sin ð Þ *πx* , and augmented with the

**Figure 1** shows that the output (54) is weakly stabilized on *ω* with error equals

**Figure 3** shows that the output (54) is weakly stabilized on *ω* with error equals

ð1 0 ð1�*<sup>t</sup>* 0

*χ* ∗

*<sup>ω</sup> χωBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup>

Then, the control

**5. Simulations**

For *ω* ¼�0*,* 2½, we have

For *ω* ¼�0*,* 3½, we have

output (54).

**Figure 1.**

**120**

*The stabilization on ω* ¼�0*,* 2½*.*

Thus

*χ* ∗

*<sup>ω</sup> χωBS t*ð Þ*z*0*; S t*ð Þ*z*<sup>0</sup> � � <sup>¼</sup> <sup>∣</sup>

weakly stabilizes the output (54).

**Figure 3.** *The stabilization on ω* ¼�0*,* 3½*.*

**Remark 2.** It is clear that the control (55) stabilizes the state on *ω*, but do not take into account the residual part Ω *ω*.

#### **6. Conclusions**

In this work, we discuss the question of output stabilization for a class of semilinear systems. Under sufficient conditions, we obtain controls depending on the output operator that strongly and weakly stabilizes the output of such systems. This work gives an opening to others questions; this is the case of output stabilization for hyperbolic semilinear systems. This will be the purpose of a future research paper.

**References**

169-179

[1] Ball JM, Slemrod M. Feedback stabilization of distributed semilinear control systems. Journal of Applied Mathematics and Optimization. 1979;**5**:

*DOI: http://dx.doi.org/10.5772/intechopen.87067*

*On the Stabilization of Infinite Dimensional Semilinear Systems*

[10] Pazy A. Semi-Groups of Linear Operators and Applications to Partial Differential Equations. New York:

[11] Lasiecka I, Tataru D. Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. Journal of Differential and Integral Equations. 1993;**6**:507-533

Springer Verlag; 1983

[2] Berrahmoune L. Stabilization and decay estimate for distributed bilinear systems. Journal of Systems Control

[3] Bounit H, Hammouri H. Feedback stabilization for a class of distributed semilinear control systems. Journal of Nonlinear Analysis. 1999;**37**:953-969

[4] Zerrik E, Ait Aadi A, Larhrissi R. Regional stabilization for a class of bilinear systems. IFAC-PapersOnLine.

[5] Zerrik E, Ait Aadi A, Larhrissi R. On the stabilization of infinite dimensional bilinear systems with unbounded control operator. Journal of Nonlinear Dynamics and Systems Theory. 2018;**18**:

[6] Zerrik E, Ait Aadi A, Larhrissi R. On the output feedback stabilization for distributed semilinear systems. Asian Journal of Control. 2019. DOI: 10.1002/

[7] Zerrik E, Ouzahra M. Regional stabilization for infinite-dimensional systems. International Journal of

[8] Zerrik E, Ouzahra M, Ztot K. Regional stabilization for infinite bilinear systems. IET Proceeding of Control Theory and Applications. 2004;

[9] Ouzahra M. Partial stabilization of semilinear systems using bounded controls. International Journal of Control. 2013;**86**:2253-2262

Control. 2003;**76**:73-81

Letters. 1999;**36**:167-171

2017;**50**:4540-4545

418-425

asjc.2081

**151**:109-116

**123**

**Figure 4.** *The evolution control in the interval* 0*,* 12*.*

#### **Author details**

El Hassan Zerrik† and Abderrahman Ait Aadi\*† MACS Team, Department of Mathematics, Moulay Ismail University, Meknes, Morocco

\*Address all correspondence to: abderrahman.aitaadi@gmail.com

† These authors contributed equally.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*On the Stabilization of Infinite Dimensional Semilinear Systems DOI: http://dx.doi.org/10.5772/intechopen.87067*

### **References**

[1] Ball JM, Slemrod M. Feedback stabilization of distributed semilinear control systems. Journal of Applied Mathematics and Optimization. 1979;**5**: 169-179

[2] Berrahmoune L. Stabilization and decay estimate for distributed bilinear systems. Journal of Systems Control Letters. 1999;**36**:167-171

[3] Bounit H, Hammouri H. Feedback stabilization for a class of distributed semilinear control systems. Journal of Nonlinear Analysis. 1999;**37**:953-969

[4] Zerrik E, Ait Aadi A, Larhrissi R. Regional stabilization for a class of bilinear systems. IFAC-PapersOnLine. 2017;**50**:4540-4545

[5] Zerrik E, Ait Aadi A, Larhrissi R. On the stabilization of infinite dimensional bilinear systems with unbounded control operator. Journal of Nonlinear Dynamics and Systems Theory. 2018;**18**: 418-425

[6] Zerrik E, Ait Aadi A, Larhrissi R. On the output feedback stabilization for distributed semilinear systems. Asian Journal of Control. 2019. DOI: 10.1002/ asjc.2081

[7] Zerrik E, Ouzahra M. Regional stabilization for infinite-dimensional systems. International Journal of Control. 2003;**76**:73-81

[8] Zerrik E, Ouzahra M, Ztot K. Regional stabilization for infinite bilinear systems. IET Proceeding of Control Theory and Applications. 2004; **151**:109-116

[9] Ouzahra M. Partial stabilization of semilinear systems using bounded controls. International Journal of Control. 2013;**86**:2253-2262

[10] Pazy A. Semi-Groups of Linear Operators and Applications to Partial Differential Equations. New York: Springer Verlag; 1983

[11] Lasiecka I, Tataru D. Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. Journal of Differential and Integral Equations. 1993;**6**:507-533

**Author details**

Morocco

**Figure 4.**

*The evolution control in the interval* 0*,* 12*.*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

†

**122**

El Hassan Zerrik† and Abderrahman Ait Aadi\*†

These authors contributed equally.

provided the original work is properly cited.

MACS Team, Department of Mathematics, Moulay Ismail University, Meknes,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: abderrahman.aitaadi@gmail.com

**Chapter 7**

**Abstract**

operator

**125**

**1. Introduction**

following form [1]:

*du t*ð Þ

8 < : *dt* ¼ �*Au t*ðÞþ <sup>ð</sup>*<sup>t</sup>*

½ � �*r;* <sup>0</sup> to *D A<sup>α</sup>* ð Þ endowed with the following norm:

0

*<sup>u</sup>*<sup>0</sup> <sup>¼</sup> *<sup>φ</sup>* <sup>∈</sup> <sup>C</sup>*<sup>α</sup>* <sup>¼</sup> *<sup>C</sup>* ½ � �*r;* <sup>0</sup> *; D A<sup>α</sup>* ð Þ ð Þ� *,*

∥*ϕ*∥*<sup>α</sup>* ¼ sup

�*r*≤ *θ* ≤ 0

Existence, Regularity, and

*α*-Norm for Some Partial

Compactness Properties in the

Functional Integrodifferential

Equations with Finite Delay

*Boubacar Diao, Khalil Ezzinbi and Mamadou Sy*

variable. An application is provided to illustrate our results.

The objective, in this work, is to study the alpha-norm, the existence, the continuity dependence in initial data, the regularity, and the compactness of solutions of mild solution for some semi-linear partial functional integrodifferential equations in abstract Banach space. Our main tools are the fractional power of linear operator theory and the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second

**Keywords:** integrodifferential, mild solution, resolvent operator, fractional power

We consider, in this manuscript, partial functional equations of retarded type with deviating arguments in terms of involving spatial partial derivatives in the

where �*A* is the infinitesimal generator of an analytic semigroup ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on a Banach space X. *B t*ð Þ is a closed linear operator with domain *DBt* ð Þ ð Þ ⊃ *D A*ð Þ timeindependent. For 0 , *α* , 1, *A<sup>α</sup>* is the fractional power of *A* which will be precise in the sequel. The domain *D A<sup>α</sup>* ð Þ is endowed with the norm <sup>∥</sup>*x*∥*<sup>α</sup>* <sup>¼</sup> <sup>∥</sup>*A<sup>α</sup>x*<sup>∥</sup> called *<sup>α</sup>*� norm. <sup>C</sup>*<sup>α</sup>* is the Banach space *<sup>C</sup>* ½ � �*r;* <sup>0</sup> *; D A<sup>α</sup>* ð Þ ð Þ of continuous functions from

*B t*ð Þ � *s u s*ð Þ*ds* þ *F t*ð Þ *; ut* for *t*≥ 0*,*

∥*ϕ θ*ð Þ∥*<sup>α</sup>* for *ϕ* ∈ C*α:*

(1)

#### **Chapter 7**

## Existence, Regularity, and Compactness Properties in the *α*-Norm for Some Partial Functional Integrodifferential Equations with Finite Delay

*Boubacar Diao, Khalil Ezzinbi and Mamadou Sy*

#### **Abstract**

The objective, in this work, is to study the alpha-norm, the existence, the continuity dependence in initial data, the regularity, and the compactness of solutions of mild solution for some semi-linear partial functional integrodifferential equations in abstract Banach space. Our main tools are the fractional power of linear operator theory and the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results.

**Keywords:** integrodifferential, mild solution, resolvent operator, fractional power operator

#### **1. Introduction**

We consider, in this manuscript, partial functional equations of retarded type with deviating arguments in terms of involving spatial partial derivatives in the following form [1]:

$$\begin{cases} \frac{du(t)}{dt} = -Au(t) + \int\_0^t B(t-s)u(s)ds + F(t, u\_t) \text{ for } t \ge 0, \\ u\_0 = \rho \in \mathcal{C}\_a = C([-r, 0], D(A^a)]), \end{cases} \tag{1}$$

where �*A* is the infinitesimal generator of an analytic semigroup ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on a Banach space X. *B t*ð Þ is a closed linear operator with domain *DBt* ð Þ ð Þ ⊃ *D A*ð Þ timeindependent. For 0 , *α* , 1, *A<sup>α</sup>* is the fractional power of *A* which will be precise in the sequel. The domain *D A<sup>α</sup>* ð Þ is endowed with the norm <sup>∥</sup>*x*∥*<sup>α</sup>* <sup>¼</sup> <sup>∥</sup>*A<sup>α</sup>x*<sup>∥</sup> called *<sup>α</sup>*� norm. <sup>C</sup>*<sup>α</sup>* is the Banach space *<sup>C</sup>* ½ � �*r;* <sup>0</sup> *; D A<sup>α</sup>* ð Þ ð Þ of continuous functions from ½ � �*r;* <sup>0</sup> to *D A<sup>α</sup>* ð Þ endowed with the following norm:

$$\|\phi\|\_a = \sup\_{-r \le \theta \le 0} \|\phi(\theta)\|\_a \text{ for } \phi \in \mathcal{C}\_{a\text{-}b}$$

*F* : R<sup>þ</sup> � C*<sup>α</sup>* ! X is a continuous function, and as usual, the history function *ut* ∈ C*<sup>α</sup>* is defined by

$$
\mu\_t(\theta) = \mu(t+\theta) \text{ for } \theta \in [-r, 0].
$$

**2. Fractional power of closed operators and resolvent operator for**

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

*<sup>A</sup>*�*<sup>α</sup>* <sup>¼</sup> <sup>1</sup>

Γð Þ¼ *α*

**Theorem 2.1.** [17] The following properties are true.

resolvent operator for the homogeneous equation of Eq. (3) if:

b.For all *x* ∈ X, *t* ! *R t*ð Þ*x* is continuous for *t*≥0.

c. *R t*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup> for *<sup>t</sup>* <sup>≥</sup>0. For *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup>, *<sup>R</sup>*ð Þ*: <sup>x</sup>* <sup>∈</sup> *<sup>C</sup>*<sup>1</sup>

*R*0

existence of an analytic resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup>.

lutely convergent for *Reλ* . 0.

a. *R*ð Þ¼ 0 *I* and k k *R t*ð Þ ≤ *M*<sup>1</sup> exp ð Þ *σt* for some *M*<sup>1</sup> ≥1 and *σ* ∈ R.

ð Þ*t x* ¼ �*AR t*ð Þ*x* þ

each *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup> and k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *b t*ð Þ∥*x*∥Y*,*for *<sup>b</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup>

¼ �*R t*ð Þ*Ax* þ

Γð Þ *α*

ð<sup>∞</sup> 0 *t α*�1

ð<sup>∞</sup> 0 *t α*�1 *e* �*t dt:*

i. <sup>Y</sup>*<sup>α</sup>* <sup>¼</sup> *D A<sup>α</sup>* ð Þ is a Banach space with the norm j j *<sup>x</sup> <sup>α</sup>* <sup>¼</sup> <sup>∥</sup>*Aαx*<sup>∥</sup> for *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup>*α*.

ii. *<sup>A</sup><sup>α</sup>* is a closed linear operator with domain <sup>Y</sup>*<sup>α</sup>* <sup>¼</sup> Im *<sup>A</sup>*�*<sup>α</sup>* ð Þ and *<sup>A</sup><sup>α</sup>* <sup>¼</sup> *<sup>A</sup>*�*<sup>α</sup>* ð Þ�<sup>1</sup>

iv. If 0 , *<sup>α</sup>* <sup>≤</sup> *<sup>β</sup>* then *D A<sup>β</sup>* � �↣*D A<sup>α</sup>* ð Þ. Moreover the injection is compact if *T t*ð Þ is

**Definition 2.2.** [18] A family of bounded linear operators ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup> in X is called

ð*t* 0

ð*t* 0

(**V1**) �*A* generates an analytic semigroup on X. ð Þ *B t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is a closed operator on X with domain at least *D A*ð Þ a.e *t*≥0 with *B t*ð Þ*x* strongly measurable for

What follows is we assume the hypothesis taken from [1] which implies the

*B t*ð Þ � *s R s*ð Þ*x ds*

*R t*ð Þ � *s B s*ð Þ*x ds:*

*T t*ð Þ*dt,*

.

(4)

ð Þ Rþ*;* X ∩*C*ð Þ Rþ*;* Y , and for *t* ≥0 we

*loc*ð Þ <sup>0</sup>*;* <sup>∞</sup> with *<sup>b</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* abso-

We shall write Y for *D A*ð Þ endowed with the graph norm k k*x* <sup>Y</sup> ¼ k k*x* þ k k *Ax ,* <sup>Y</sup>*<sup>α</sup>* for *D A<sup>α</sup>* ð Þ and <sup>L</sup>ð Þ <sup>Y</sup>*α;* <sup>X</sup> will denote the space of bounded linear operators from <sup>Y</sup>*<sup>α</sup>* to <sup>X</sup>, and for <sup>Y</sup><sup>0</sup> <sup>¼</sup> <sup>X</sup>, we write <sup>L</sup>ð Þ <sup>X</sup> with norm k k*:* <sup>L</sup>ð Þ <sup>X</sup> . We also frequently use the Laplace transform of *<sup>f</sup>* which is denoted by *<sup>f</sup>* <sup>∗</sup> . If we assume that �*<sup>A</sup>* generates an analytic semigroup and, without loss of generality, that 0 ∈ *ϱ*ð Þ *A* , then one can define the fractional power *A<sup>α</sup>* for 0 , *α* , 1, as a closed linear operator on its

**integrodifferential equations**

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

domain Y*<sup>α</sup>* with its inverse *A*�*<sup>α</sup>* given by

where Γ is the gamma function

We have the following known results.

iii. *A*�*<sup>α</sup>* is a bounded linear operator in X.

compact for *t* . 0.

have

**127**

As a model for this class, one may take the following Lotka-Volterra equation:

$$\begin{cases} \frac{\partial u(t, \mathbf{x})}{\partial t} = \frac{\partial^2 u(t, \mathbf{x})}{\partial \mathbf{x}^2} + \int\_0^t h(t - s) \frac{\partial^2 u(s, \mathbf{x})}{\partial \mathbf{x}^2} ds \\\\ \qquad + \int\_{-r}^0 g\left(t, \frac{\partial u(t + \theta, \mathbf{x})}{\partial \mathbf{x}}\right) d\theta \text{ for } t \ge 0 \text{ and } \mathbf{x} \in [0, \pi], \\\\ u(t, 0) = u(t, \pi) = 0 \text{ for } t \ge 0, \\\\ u(\theta, \mathbf{x}) = u\_0(\theta, \mathbf{x}) \text{ for } \theta \in [-r, 0] \text{ and } \boldsymbol{x} \in [0, \pi]. \end{cases} \tag{2}$$

Here *u*<sup>0</sup> : ½ �� �*r;* 0 ½ �! 0*; π* R*, g* : R<sup>þ</sup> � R ! R and *h* : R<sup>þ</sup> ! R are appropriate functions.

In the particular case where *α* ¼ 0, many results are obtained in the literature under various hypotheses concerning *A*, *B*, and *F* (see, for instance, [2–6] and the references therein). For example, in [7], Ezzinbi et al. investigated the existence and regularity of solutions of the following equation:

$$\begin{cases} \frac{du(t)}{dt} = -Au(t) + \int\_0^t B(t-s)u(s)ds + F(t, u\_t) \text{ for } t \ge 0, \\ u\_0 = \rho \in \mathcal{C}([-r, 0]; \mathbb{X}), \end{cases} \tag{3}$$

The authors obtained also the uniqueness and the representation of solutions via a variation of constant formula, and other properties of the resolvent operator were studied. In [8], Ezzinbi et al. studied a local existence and regularity of Eq. (3). To achieve their goal, the authors used the variation of constant formula, the theory of resolvent operator, and the principle contraction method. Ezzinbi et al. in [9] studied the local existence and global continuation for Eq. (3). Recall that the resolvent operator plays an important role in solving Eq. (3); in the weak and strict sense, it replaces the role of the *c*<sup>0</sup> semigroup theory. For more details in this topic, here are the papers of Chen and Grimmer [2], Hannsgen [10], Smart [11], Miller [12, 13], and Miller and Wheeler [14, 15]. In the case where the nonlinear part involves spatial derivative, the above obtained results become invalid. To overcome this difficulty, we shall restrict our problem in a Banach space Y*<sup>α</sup>* ⊂ X, to obtain our main results for Eq. (1).

Considering the case where *B* ¼ 0, Travis and Webb in [16] obtained results on the existence, stability, regularity, and compactness of Eq. (1). To achieve their goal, the authors assumed that �*A* is the infinitesimal generator of a compact analytic semigroup and *F* is only continuous with respect to a fractional power of *A* in the second variable. The present paper is motivated by the paper of Travis and Webb in [16].

The paper is organized as follows. In Section 2, we recall some fundamental properties of the resolvent operator and fractional powers of closed operators. The global existence, uniqueness, and continuous dependence with respect to the initial data are studied in Section 3. In Section 4, we study the local existence and bowing up phenomena. In Section 5 we prove, under some conditions, the regularity of the mild solutions. And finally, we illustrate our main results in Section 6 by examining an example.

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

#### **2. Fractional power of closed operators and resolvent operator for integrodifferential equations**

We shall write Y for *D A*ð Þ endowed with the graph norm k k*x* <sup>Y</sup> ¼ k k*x* þ k k *Ax ,* <sup>Y</sup>*<sup>α</sup>* for *D A<sup>α</sup>* ð Þ and <sup>L</sup>ð Þ <sup>Y</sup>*α;* <sup>X</sup> will denote the space of bounded linear operators from <sup>Y</sup>*<sup>α</sup>* to <sup>X</sup>, and for <sup>Y</sup><sup>0</sup> <sup>¼</sup> <sup>X</sup>, we write <sup>L</sup>ð Þ <sup>X</sup> with norm k k*:* <sup>L</sup>ð Þ <sup>X</sup> . We also frequently use the Laplace transform of *<sup>f</sup>* which is denoted by *<sup>f</sup>* <sup>∗</sup> . If we assume that �*<sup>A</sup>* generates an analytic semigroup and, without loss of generality, that 0 ∈ *ϱ*ð Þ *A* , then one can define the fractional power *A<sup>α</sup>* for 0 , *α* , 1, as a closed linear operator on its domain Y*<sup>α</sup>* with its inverse *A*�*<sup>α</sup>* given by

$$A^{-a} = \frac{1}{\Gamma(a)} \int\_0^\infty t^{a-1} T(t) \, dt,$$

where Γ is the gamma function

*F* : R<sup>þ</sup> � C*<sup>α</sup>* ! X is a continuous function, and as usual, the history function

*ut*ð Þ¼ *θ u t*ð Þ þ *θ* for *θ* ∈ ½ � �*r;* 0 *:*

As a model for this class, one may take the following Lotka-Volterra equation:

Here *u*<sup>0</sup> : ½ �� �*r;* 0 ½ �! 0*; π* R*, g* : R<sup>þ</sup> � R ! R and *h* : R<sup>þ</sup> ! R are appropriate

In the particular case where *α* ¼ 0, many results are obtained in the literature under various hypotheses concerning *A*, *B*, and *F* (see, for instance, [2–6] and the references therein). For example, in [7], Ezzinbi et al. investigated the existence and

The authors obtained also the uniqueness and the representation of solutions via a variation of constant formula, and other properties of the resolvent operator were studied. In [8], Ezzinbi et al. studied a local existence and regularity of Eq. (3). To achieve their goal, the authors used the variation of constant formula, the theory of resolvent operator, and the principle contraction method. Ezzinbi et al. in [9] studied the local existence and global continuation for Eq. (3). Recall that the resolvent operator plays an important role in solving Eq. (3); in the weak and strict sense, it replaces the role of the *c*<sup>0</sup> semigroup theory. For more details in this topic, here are the papers of Chen and Grimmer [2], Hannsgen [10], Smart [11], Miller [12, 13], and Miller and Wheeler [14, 15]. In the case where the nonlinear part involves spatial derivative, the above obtained results become invalid. To overcome this difficulty, we shall restrict our problem in a Banach space Y*<sup>α</sup>* ⊂ X, to obtain our

Considering the case where *B* ¼ 0, Travis and Webb in [16] obtained results on the existence, stability, regularity, and compactness of Eq. (1). To achieve their goal, the authors assumed that �*A* is the infinitesimal generator of a compact analytic semigroup and *F* is only continuous with respect to a fractional power of *A* in the second variable. The present paper is motivated by the paper of Travis and Webb in [16]. The paper is organized as follows. In Section 2, we recall some fundamental properties of the resolvent operator and fractional powers of closed operators. The global existence, uniqueness, and continuous dependence with respect to the initial data are studied in Section 3. In Section 4, we study the local existence and bowing up phenomena. In Section 5 we prove, under some conditions, the regularity of the mild solutions. And finally, we illustrate our main results in Section 6 by examining

*dθ* for *t*≥0 and *x* ∈ ½ � 0*; π ,*

*B t*ð Þ � *s u s*ð Þ*ds* þ *F t*ð Þ *; ut* for *t*≥ 0*,*

(2)

(3)

*∂*2 *u s*ð Þ *; x <sup>∂</sup>x*<sup>2</sup> *ds*

*ut* ∈ C*<sup>α</sup>* is defined by

*<sup>∂</sup>u t*ð Þ *; <sup>x</sup> <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

8

>>>>>>>>>>>>><

>>>>>>>>>>>>>:

functions.

*u t*ð Þ *; x ∂x*<sup>2</sup> þ

*u t*ð Þ¼ *;* 0 *u t*ð Þ¼ *; π* 0 for *t*≥0*,*

regularity of solutions of the following equation:

*dt* ¼ �*Au t*ðÞþ

*u*<sup>0</sup> ¼ *φ* ∈ *C*ð Þ ½ � �*r;* 0 ; X *,*

*du t*ð Þ

8 < :

main results for Eq. (1).

an example.

**126**

þ ð0 �*r g t;* ð*t* 0

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*h t*ð Þ � *s*

*<sup>∂</sup>u t*ð Þ <sup>þ</sup> *<sup>θ</sup>; <sup>x</sup> ∂x* � �

*u*ð Þ¼ *θ; x u*0ð Þ *θ; x* for *θ* ∈ ½ � �*r;* 0 and *x* ∈ ½ � 0*; π :*

ð*t* 0

$$\Gamma(a) = \int\_0^\infty t^{a-1} e^{-t} \, dt.$$

We have the following known results. **Theorem 2.1.** [17] The following properties are true.


**Definition 2.2.** [18] A family of bounded linear operators ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup> in X is called resolvent operator for the homogeneous equation of Eq. (3) if:


$$\begin{split} R'(t)\mathbf{x} &= -AR(t)\mathbf{x} + \int\_0^t B(t-s)R(s)\mathbf{x} \, ds \\ &= -R(t)A\mathbf{x} + \int\_0^t R(t-s)B(s)\mathbf{x} \, ds. \end{split} \tag{4}$$

What follows is we assume the hypothesis taken from [1] which implies the existence of an analytic resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup>.

(**V1**) �*A* generates an analytic semigroup on X. ð Þ *B t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is a closed operator on X with domain at least *D A*ð Þ a.e *t*≥0 with *B t*ð Þ*x* strongly measurable for each *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup> and k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *b t*ð Þ∥*x*∥Y*,*for *<sup>b</sup>* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> *loc*ð Þ <sup>0</sup>*;* <sup>∞</sup> with *<sup>b</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* absolutely convergent for *Reλ* . 0.

(**V2**) *ρ λ*ð Þ¼ *<sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � *<sup>B</sup>*<sup>∗</sup> ð Þ ð Þ*<sup>λ</sup>* �<sup>1</sup> exists as a bounded operator on <sup>X</sup> which is analytic for *λ* ∈ Λ ¼ f g *λ* ∈ C : j j *argλ* , *π=*2 þ *δ ,* where 0 , *δ* , *π=*2. In Λ if j j *λ* ≥ ϵ . 0, there exists *M* ¼ *M*ð Þϵ . 0 so that k k *ρ λ*ð Þ ≤ *M=*j j *λ* .

∥*y*∥*<sup>α</sup>* ¼ sup

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

Γ*y t*ðÞ¼ *R t*ð Þ*φ*ð Þþ 0

ð Þ Γð Þ� *u* Γð Þ*v* ðÞ¼ *t*

*<sup>A</sup><sup>α</sup>* k k ð Þ <sup>Γ</sup>ð Þ� *<sup>u</sup>* <sup>Γ</sup>ð Þ*<sup>v</sup>* ð Þ*<sup>t</sup>* <sup>≤</sup> *LFN<sup>α</sup>*

Using the *α*� norm, we have

Now we choose *a* such that

*dt z t*ðÞ¼�*Az t*ðÞþ <sup>ð</sup>*<sup>t</sup>*

*za* ¼ *ya* ∈ *C*ð Þ ½ � �*r; a ;* Y*<sup>α</sup> :*

*d*

8 < :

**129**

0≤ *t*≤ *a*

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

ð*t* 0

> ð*t* 0

≤ *LFN<sup>α</sup>*

≤ *LFN<sup>α</sup>*

*LFN<sup>α</sup>* ð*a* 0 *ds <sup>s</sup><sup>α</sup>* , <sup>1</sup>*:*

*a*

Let *z* be the function defined by *z t*ðÞ¼ *z t*ð Þ for *t* ∈ ½ � *a;* 2*a* and *z t*ðÞ¼ *y t*ð Þ for

∧ ¼ f g *z* ∈ *C a* ð Þ ½ � *;* 2*a* ; Y*<sup>α</sup>* : *z a*ð Þ¼ *y a*ð Þ *,*

*a*

provided with the induced topological norm. We define the operator Γ*<sup>a</sup>* on ∧ by

we show that the following equation has a unique mild solution:

*a*

Notice that the solution of Eq. (6) is given by

*z t*ðÞ¼ *R t*ð Þ � *<sup>a</sup> z a*ð Þþ <sup>ð</sup>*<sup>t</sup>*

*t* ∈ ½ � �*r; a* . Consider now again the set ∧ defined by

ð Þ <sup>Γ</sup>*az* ðÞ¼ *<sup>t</sup> R t*ð Þ � *<sup>a</sup> z a*ð Þþ <sup>ð</sup>*<sup>t</sup>*

ð*t* 0

ð*t* 0

> ð*a* 0 *ds sα*

Then Γ is a strict contraction on ∧, and it has a unique fixed point *y* which is the unique mild solution of Eq. (1) on 0½ � *; a* . To extend the solution of Eq. (1) in ½ � *a;* 2*a* ,

*B t*ð Þ � *s z s*ð Þ*ds* þ *F t*ð Þ *; zt* for *t* ∈ ½ � *a;* 2*a ,*

*R t*ð Þ � *s F s*ð Þ *; zs ds* for *t* ∈ ½ � *a;* 2*a :*

*R t*ð Þ � *s F s*ð Þ *; zs ds* for *t* ∈ ½ � *a;* 2*a :*

� �

1

1 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* sup 0≤ *τ* ≤ *a*

∥*y t*ð Þ∥*<sup>α</sup>* for *y* ∈ *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup> :*

� � *ds* for *<sup>t</sup>* <sup>∈</sup> ½ � <sup>0</sup>*; <sup>a</sup> :*

*R t*ð Þ � *s* ð Þ *F s*ð Þ� *; us F s*ð Þ *; vs ds:*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* k k *us* � *vs <sup>α</sup> ds*

k k *u* � *v <sup>α</sup>:*

k k *u*ð Þ� *τ v*ð Þ*τ <sup>α</sup> ds*

(6)

For *y* ∈ ∧, we introduce the extension *y* of *y* on ½ � �*r; a* defined by *y t*ðÞ¼ *y t*ð Þ for *t* ∈ ½ � 0*; a* and *y t*ðÞ¼ *φ*ð Þ*t* for *t* ∈ ½ � �*r;* 0 . We consider the operator Γ defined on ∧ by

*R t*ð Þ � *s F s; ys*

We claim that Γð Þ ∧ ⊂ ∧*:* In fact for *y* ∈ ∧, we have ð Þ Γ*y* ð Þ¼ 0 *φ*ð Þ 0 , and by continuity of *F* and *R t*ð Þ*x* for *x* ∈ X, we deduce that Γ*y* ∈ ∧*:* In order to obtain our result, we apply the strict contraction principle. In fact, let *u, v* ∈ ∧ and *t* ∈ ½ � 0*; a* . Then

(**V3**) *<sup>A</sup>ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>X</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> and is analytic from <sup>Λ</sup> to <sup>L</sup>ð Þ <sup>X</sup> . *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Given <sup>ϵ</sup> . 0, there exists a positive constant *<sup>M</sup>* <sup>¼</sup> *<sup>M</sup>*ð Þ*<sup>ε</sup>* so that k k *<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>þ</sup> *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup> ρ λ*ð Þ*<sup>x</sup>* <sup>≤</sup> ð Þ *<sup>M</sup>=*j j *<sup>λ</sup>* k k*<sup>x</sup>* <sup>Y</sup> for *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup> and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> with *<sup>λ</sup>*<sup>≥</sup> *<sup>ε</sup>* and *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup>* <sup>↦</sup>0 as j j *<sup>λ</sup>* <sup>↦</sup><sup>∞</sup> in <sup>Λ</sup>. In addition, k k *<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>M</sup>*j j *<sup>λ</sup> <sup>n</sup>* ∥*x*∥ for some *n* . 0*, λ* ∈ Λ with *λ*≥*ε*. Further, there exists *<sup>D</sup>* <sup>⊂</sup> *D A*<sup>2</sup> � � which is dense in <sup>Y</sup> such that *A D*ð Þ and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* ð Þ *<sup>D</sup>* are contained in <sup>Y</sup> and *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup> <sup>x</sup>* <sup>Y</sup> is bounded for each *<sup>x</sup>* <sup>∈</sup> *<sup>D</sup>* and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> with j j *<sup>λ</sup>* <sup>≥</sup>ϵ.

**Theorem 2.3.** [1] Assume that conditions (**V1**)–(**V3**) are satisfied. Then there exists an analytic resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥0. Moreover, there exist positive constants *N, N<sup>α</sup>* such that k k *R t*ð Þ <sup>≤</sup> *<sup>N</sup>*and *<sup>A</sup><sup>α</sup>* k k *R t*ð Þ <sup>≤</sup> *<sup>N</sup><sup>α</sup> <sup>t</sup><sup>α</sup>* for *t* . 0 and 0 ≤ *α* , 1.

We take the following hypothesis.

(**H0**) The semigroup ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is compact for *t* . 0.

**Theorem 2.4.** [19] Under the conditions (**V1**)–(**V3**) and (**H0**), the corresponding resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is compact for *t* . 0.

#### **3. Global existence, uniqueness, and continuous dependence with respect to the initial data**

**Definition 3.1.** A function *u* : ½ �! 0*; b* Y*<sup>α</sup>* is called a strict solution of Eq. (1), if:

i. *t* ! *u t*ð Þ is continuously differentiable on 0½ � *; b* .

ii. *u t*ð Þ ∈ Y for *t* ∈ ½ � 0*; b* .

iii. *u* satisfies Eq. (1) on 0½ � *; b* .

**Definition 3.2.** A continuous function *u* : ½ �! 0*; b* Y*<sup>α</sup>* is called a mild solution of Eq. (1) if

$$\begin{cases} u(t) = R(t)\rho(\mathbf{0}) + \int\_0^t R(t-s)F(s,u\_s)ds \text{ for } t \in [0,b],\\ u\_0 = \boldsymbol{\rho} \in \mathcal{C}\_a. \end{cases} \tag{5}$$

Now to obtain our first result, we take the following assumption. (**H1**) There exists a constant *LF* . 0 such that

$$\|\|F(t,\rho\_1) - F(t,\rho\_2)\|\| \le L\_F \|\|\rho\_1 - \rho\_2\|\|\_a \text{ for } t \ge 0 \text{ and } \rho\_1, \rho\_2 \in \mathcal{C}\_a.$$

**Theorem 3.3.** Assume that (**V1**)–(**V3**) and (**H1**) hold. Then for *φ* ∈ C*α*, Eq. (1) has a unique mild solution which is defined for all *t*≥ 0*:*

*Proof.* Let *a* . 0. For *φ* ∈ C*α*, we define the set ∧ by

$$\wedge = \{ \mathcal{Y} \in \mathcal{C}([\mathbf{0}, a]; \mathbb{Y}\_a) : \mathcal{Y}(\mathbf{0}) = \boldsymbol{\varrho}(\mathbf{0}) \}.$$

The set ∧ is a closed subset of *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup>* where *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup>* is the space of continuous functions from 0½ � *; a* to Y*<sup>α</sup>* equipped with the uniform norm topology *Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

$$\|\boldsymbol{y}\|\_{a} = \sup\_{0 \le t \le a} \|\boldsymbol{y}(t)\|\_{a} \text{ for } \boldsymbol{y} \in C([0, a]; \mathbb{Y}\_{a}).$$

For *y* ∈ ∧, we introduce the extension *y* of *y* on ½ � �*r; a* defined by *y t*ðÞ¼ *y t*ð Þ for *t* ∈ ½ � 0*; a* and *y t*ðÞ¼ *φ*ð Þ*t* for *t* ∈ ½ � �*r;* 0 . We consider the operator Γ defined on ∧ by

$$
\Gamma \mathfrak{y}(t) = R(t)\mathfrak{q}(\mathbf{0}) + \int\_0^t R(t-s)F(s, \overline{\mathfrak{y}}\_s) \, ds \text{ for } t \in [0, a].
$$

We claim that Γð Þ ∧ ⊂ ∧*:* In fact for *y* ∈ ∧, we have ð Þ Γ*y* ð Þ¼ 0 *φ*ð Þ 0 , and by continuity of *F* and *R t*ð Þ*x* for *x* ∈ X, we deduce that Γ*y* ∈ ∧*:* In order to obtain our result, we apply the strict contraction principle. In fact, let *u, v* ∈ ∧ and *t* ∈ ½ � 0*; a* . Then

$$(\Gamma(u) - \Gamma(v))(t) = \int\_0^t R(t - s)(F(s, \overline{u}\_s) - F(s, \overline{v}\_s)) \, ds.$$

Using the *α*� norm, we have

(**V2**) *ρ λ*ð Þ¼ *<sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � *<sup>B</sup>*<sup>∗</sup> ð Þ ð Þ*<sup>λ</sup>* �<sup>1</sup> exists as a bounded operator on <sup>X</sup> which is

(**V3**) *<sup>A</sup>ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>X</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> and is analytic from <sup>Λ</sup> to <sup>L</sup>ð Þ <sup>X</sup> . *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Given <sup>ϵ</sup> . 0, there exists a positive constant *<sup>M</sup>* <sup>¼</sup> *<sup>M</sup>*ð Þ*<sup>ε</sup>* so that k k *<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>þ</sup> *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup> ρ λ*ð Þ*<sup>x</sup>* <sup>≤</sup> ð Þ *<sup>M</sup>=*j j *<sup>λ</sup>* k k*<sup>x</sup>* <sup>Y</sup> for *<sup>x</sup>* <sup>∈</sup> <sup>Y</sup> and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> with *<sup>λ</sup>*<sup>≥</sup> *<sup>ε</sup>* and *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup>* <sup>↦</sup>0 as j j *<sup>λ</sup>* <sup>↦</sup><sup>∞</sup> in <sup>Λ</sup>. In addition,

**Theorem 2.3.** [1] Assume that conditions (**V1**)–(**V3**) are satisfied. Then there exists an analytic resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥0. Moreover, there exist positive con-

j j *λ* ≥ ϵ . 0, there exists *M* ¼ *M*ð Þϵ . 0 so that k k *ρ λ*ð Þ ≤ *M=*j j *λ* .

k k *<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>M</sup>*j j *<sup>λ</sup> <sup>n</sup>*

stants *N, N<sup>α</sup>* such that k k *R t*ð Þ <sup>≤</sup> *<sup>N</sup>*and *<sup>A</sup><sup>α</sup>* k k *R t*ð Þ <sup>≤</sup> *<sup>N</sup><sup>α</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

(**H0**) The semigroup ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is compact for *t* . 0.

i. *t* ! *u t*ð Þ is continuously differentiable on 0½ � *; b* .

*u t*ðÞ¼ *R t*ð Þ*φ*ð Þþ 0

(**H1**) There exists a constant *LF* . 0 such that

has a unique mild solution which is defined for all *t*≥ 0*: Proof.* Let *a* . 0. For *φ* ∈ C*α*, we define the set ∧ by

*u*<sup>0</sup> ¼ *φ* ∈ C*α:*

corresponding resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup> is compact for *t* . 0.

**Theorem 2.4.** [19] Under the conditions (**V1**)–(**V3**) and (**H0**), the

**3. Global existence, uniqueness, and continuous dependence with**

**Definition 3.1.** A function *u* : ½ �! 0*; b* Y*<sup>α</sup>* is called a strict solution of Eq. (1), if:

**Definition 3.2.** A continuous function *u* : ½ �! 0*; b* Y*<sup>α</sup>* is called a mild solution

*F t; φ*<sup>1</sup> ð Þ� *F t; φ*<sup>2</sup> k k ð Þ ≤ *LF*k k *φ*<sup>1</sup> � *φ*<sup>2</sup> *<sup>α</sup>* for *t*≥ 0 and *φ*1*, φ*<sup>2</sup> ∈ C*α:*

**Theorem 3.3.** Assume that (**V1**)–(**V3**) and (**H1**) hold. Then for *φ* ∈ C*α*, Eq. (1)

∧ ¼ f g *y* ∈ *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup>* : *y*ð Þ¼ 0 *φ*ð Þ 0 *:*

The set ∧ is a closed subset of *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup>* where *C*ð Þ ½ � 0*; a* ; Y*<sup>α</sup>* is the space of continuous functions from 0½ � *; a* to Y*<sup>α</sup>* equipped with the uniform norm topology

*R t*ð Þ � *s F s*ð Þ *; us ds* for *t* ∈ ½ � 0*; b ,*

(5)

ð*t* 0

Now to obtain our first result, we take the following assumption.

We take the following hypothesis.

**respect to the initial data**

ii. *u t*ð Þ ∈ Y for *t* ∈ ½ � 0*; b* .

8 < :

of Eq. (1) if

**128**

iii. *u* satisfies Eq. (1) on 0½ � *; b* .

analytic for *λ* ∈ Λ ¼ f g *λ* ∈ C : j j *argλ* , *π=*2 þ *δ ,* where 0 , *δ* , *π=*2. In Λ if

*<sup>D</sup>* <sup>⊂</sup> *D A*<sup>2</sup> � � which is dense in <sup>Y</sup> such that *A D*ð Þ and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* ð Þ *<sup>D</sup>* are contained in <sup>Y</sup> and *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup> <sup>x</sup>* <sup>Y</sup> is bounded for each *<sup>x</sup>* <sup>∈</sup> *<sup>D</sup>* and *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> with j j *<sup>λ</sup>* <sup>≥</sup>ϵ.

∥*x*∥ for some *n* . 0*, λ* ∈ Λ with *λ*≥*ε*. Further, there exists

*<sup>t</sup><sup>α</sup>* for *t* . 0 and 0 ≤ *α* , 1.

$$\begin{split} \left||A^{a}(\Gamma(u)-\Gamma(v))(t)|| \leq L\_{F}N\_{a} \int\_{0}^{t} \frac{1}{\left(t-s\right)^{a}} \left||\overline{u}\_{s}-\overline{v}\_{i}\right||\_{a} ds \\ \leq L\_{F}N\_{a} \int\_{0}^{t} \frac{1}{\left(t-s\right)^{a}} \sup\_{0 \leq \tau \leq a} ||u(\tau)-v(\tau)||\_{a} \, ds \\ \leq \left(L\_{F}N\_{a} \int\_{0}^{t} \frac{ds}{s^{a}}\right) \left||u-v\right||\_{a} .\end{split}$$

Now we choose *a* such that

$$L\_F \mathcal{N}\_a \int\_0^a \frac{ds}{s^a} < 1.$$

Then Γ is a strict contraction on ∧, and it has a unique fixed point *y* which is the unique mild solution of Eq. (1) on 0½ � *; a* . To extend the solution of Eq. (1) in ½ � *a;* 2*a* , we show that the following equation has a unique mild solution:

$$\begin{cases} \frac{d}{dt}z(t) = -Az(t) + \int\_{a}^{t} B(t-s)z(s)ds + F(t, z\_t) \text{ for } t \in [a, 2a],\\ z\_a = y\_a \in \mathcal{C}([-r, a], \mathbb{Y}\_a). \end{cases} \tag{6}$$

Notice that the solution of Eq. (6) is given by

$$x(t) = R(t - a)x(a) + \int\_{a}^{t} R(t - s)F(s, x\_s) \, ds \text{ for } t \in [a, 2a].$$

Let *z* be the function defined by *z t*ðÞ¼ *z t*ð Þ for *t* ∈ ½ � *a;* 2*a* and *z t*ðÞ¼ *y t*ð Þ for *t* ∈ ½ � �*r; a* . Consider now again the set ∧ defined by

$$\wedge = \{ z \in \mathcal{C}([a, 2a]; \mathbb{Y}\_a) : z(a) = \mathfrak{y}(a) \},$$

provided with the induced topological norm. We define the operator Γ*<sup>a</sup>* on ∧ by

$$\mathcal{R}(\Gamma\_{\mathfrak{a}}\mathbf{z})(t) = \mathcal{R}(t - \mathfrak{a})\mathbf{z}(\mathfrak{a}) + \int\_{\mathfrak{a}}^{t} \mathcal{R}(t - \mathfrak{s})\mathcal{F}(\mathfrak{s}, \overline{\mathfrak{z}}\_{\mathfrak{s}}) \, d\mathfrak{s} \text{ for } t \in [\mathfrak{a}, 2\mathfrak{a}].$$

We have ð Þ Γ*az* ð Þ¼ *a y a*ð Þ and Γ*az* is continuous. Then it follows that Γ*a*∧⊂∧*:* Moreover, for *u, v* ∈ ∧, one has

$$||A^a(\Gamma\_a(u) - \Gamma\_a(v))(t)|| \le L\_F N\_a \int\_a^t \frac{1}{(t-s)^a} \left\| \overline{u}\_s - \overline{v}\_s \right\|\_a ds.$$

Since *u* ¼ *v* ¼ *φ* in ½ � �*r;* 0 *,* we deduce that

$$||A^a(\Gamma\_a(u) - \Gamma\_a(v))|| \le \left(L\_F N\_a \int\_0^a \frac{ds}{s^a} \right) ||u - v||\_a.$$

Then we deduce that Γ*<sup>a</sup>* has a unique fixed point in ∧ which extends the solution *y* in ½ � *a;* 2*a* . Proceeding inductively, *y* is uniquely and continuously extended to ½ � *na;*ð Þ *n* þ 1 *a* for all *n*≥ 1, and this ends the proof.

Now we show the continuous dependence of the mild solutions with respect to the initial data.

**Theorem 3.4.** Assume that (**V1**)–(**V3**) and (**H1**) hold. Then the mild solution *u*ð Þ *:; φ* of Eq. (1) defines a continuous Lipschitz operator *U t*ð Þ*, t*≥0 in C*<sup>α</sup>* by *U t*ð Þ*φ* ¼ *ut*ð Þ *:; φ* . That is, *U t*ð Þ*φ* is continuous from 0½ Þ ; ∞ to C*<sup>α</sup>* for each fixed *φ* ∈ C*α*. Moreover there exist a real number *δ* and a scalar function *P* such that for *t*≥0 and *φ*1*, φ*<sup>2</sup> ∈ C*<sup>α</sup>* we have

$$\|U(t)\varphi\_1 - U(t)\varphi\_2\| \le P(\delta)e^{\delta t} \|\varphi\_1 - \varphi\_2\|\_a. \tag{7}$$

If �*r*≤ *τ* ≤ 0, we have

For 0 ≤ *t*≤ *t*, we have

sup�*r*<sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup> <sup>e</sup>*

*e* �*δt*

**the flow**

*x* ∈ X*β,* with *b* ∈ **L***<sup>q</sup>*

**131**

which implies that

Then the result follows.

*e*

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

∥*wt*∥*<sup>α</sup>* ¼ sup

Then from Eqs. (11) and (12), we deduce that for 0 ≤ *t*≤ *t*

�*r*≤ *θ* ≤ 0

≤ *M*<sup>4</sup> sup �*r*≤ *θ* ≤ 0

≤ *M*<sup>4</sup> sup �*r*≤ *τ* ≤ *t*

Therefore, Eqs. (9) and (10) imply that

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

*e* �*δt* �*δτ*∥*w*ð Þ*<sup>τ</sup>* <sup>∥</sup>*<sup>α</sup>* ≤ ∥*φ*<sup>1</sup> � *<sup>φ</sup>*2∥*αM*3*:* (10)

�*δ*ð Þ *<sup>t</sup>*þ*<sup>θ</sup>* <sup>∥</sup>*w t*ð Þ <sup>þ</sup> *<sup>θ</sup>* <sup>∥</sup>*<sup>α</sup>*

�*δ*ð Þ *<sup>t</sup>*þ*<sup>θ</sup>* <sup>∥</sup>*w t*ð Þ <sup>þ</sup> *<sup>θ</sup>* <sup>∥</sup>*<sup>α</sup>*

�*δτ*∥*w*ð Þ*<sup>τ</sup>* <sup>∥</sup>*α:*

*:* (11)

(12)

*,*

*:*

�*δτ*∥*w*ð Þ*<sup>τ</sup>* <sup>∥</sup>*<sup>α</sup>* <sup>≤</sup> *<sup>M</sup>*1*M*3∥*φ*<sup>1</sup> � *<sup>φ</sup>*2∥<sup>C</sup>*<sup>α</sup>* <sup>þ</sup> *LFM*1*E*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup>

*e*

*e*

<sup>∥</sup>*wt*∥*<sup>α</sup>* <sup>≤</sup> *<sup>M</sup>*1*M*3*M*4∥*φ*<sup>1</sup> � *<sup>φ</sup>*2∥*<sup>α</sup>* <sup>þ</sup> *LFM*1*M*3*E*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup>

*<sup>E</sup>*<sup>≤</sup> *<sup>M</sup>*1*M*3*M*4∥*φ*<sup>1</sup> � *<sup>φ</sup>*2∥*<sup>α</sup>* <sup>þ</sup> *LFM*1*M*4*E*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup>

**4. Local existence, blowing up phenomena, and the compactness of**

on X ð Þ *α* ¼ 0 , in the case where *α* 6¼ 0. We take the following assumption.

*loc*ð Þ 0*;* ∞ where *q* . 1*=*ð Þ 1 � *β :*

exists a positive constant *M* ¼ *M a*ð Þ such that for *x* ∈ X we have

*<sup>A</sup><sup>α</sup>* k k ð Þ *R t*ð Þ <sup>þ</sup> *<sup>h</sup> <sup>x</sup>* � *R h*ð Þ*R t*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>M</sup>*

*dt R t*ð Þ <sup>þ</sup> *<sup>h</sup> <sup>x</sup>* ¼ �*AR t*ð Þ <sup>þ</sup> *<sup>h</sup> <sup>x</sup>* <sup>þ</sup>

þ ð*<sup>t</sup>*þ*<sup>h</sup> t*

We deduce that *R t*ð Þ þ *h x* satisfies the equation of the form

¼ �*AR t*ð Þ þ *h x* þ

*Proof.* Let *a* . 0 and *x* ∈ X. Then

*d*

We start by generalizing a result, obtained in [19] in the case of the usual norm

**Theorem 4.1.** Assume that (**V1**)–(**V3**) and (**H2**) hold. Then for any *a* . 0*,* there

ð*h* 0 *ds*

ð*<sup>t</sup>*þ*<sup>h</sup>* 0

ð*t* 0

*B t*ð Þ þ *h* � *s R s*ð Þ*x ds:*

*<sup>s</sup><sup>α</sup>* k k*<sup>x</sup>* for <sup>0</sup><sup>≤</sup> *<sup>h</sup>* , *<sup>t</sup>*<sup>≤</sup> *<sup>a</sup>:*

*B t*ð Þ þ *h* � *s R s*ð Þ*x ds*

*B t*ð Þ � *s R s*ð Þ þ *h x ds*

(**H2**) *B t*ð Þ <sup>∈</sup> <sup>L</sup> <sup>X</sup>*β;* <sup>X</sup> � � for some 0 , *<sup>β</sup>* , 1, a.e *<sup>t</sup>*≥0 and k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *b t*ð Þk k*<sup>x</sup> <sup>β</sup>* for

*e δθe*

*Proof.* We use the gamma formula

$$
\Gamma(1-a)k^{a-1} = \int\_0^\infty e^{-ks}s^{-a}ds,
$$

where *k* . 0 (see [20], p. 265). The continuity is obvious that the map *t* ! *ut*ð Þ *:; φ* is continuous. Now, let *φ*1*, φ*<sup>2</sup> ∈ C*α:* If we pose *w t*ðÞ¼ *u t; φ*<sup>1</sup> ð Þ� *u t; φ*<sup>2</sup> ð Þ*,* then we have

$$\|\boldsymbol{w}(t)\|\_{a} \leq M\_1 e^{\sigma t} \|\boldsymbol{\varrho}\_1 - \boldsymbol{\varrho}\_2\|\_{a} + L\_F N\_a \int\_0^t \frac{e^{\sigma(t-s)}}{(t-s)^a} \|\boldsymbol{w}\_s\|\_a ds. \tag{8}$$

Let *δ* a real number be such that

$$
\sigma - \delta \le 0 \text{ and } M\_1 \max\{e^{-\delta r}, \mathbf{1}\} L\_F \Gamma(\mathbf{1} - a)(\delta - \sigma)^{a-1} \le \mathbf{1}.
$$

We define the function *P* by

$$P(\delta) = M\_1 M\_3 M\_4 \left( \mathbf{1} - M\_1 M\_4 L\_F \Gamma (\mathbf{1} - a) (\delta - \sigma)^{a-1} \right)^{-1}$$

where

$$M\_3 = \max\{e^{\delta r}, \mathbf{1}\}, M\_4 = \max\{e^{-\delta r}, \mathbf{1}\}.$$

Fix *<sup>t</sup>* . 0 and let *<sup>E</sup>* <sup>¼</sup> sup0 <sup>≤</sup> *<sup>s</sup>*<sup>≤</sup> *<sup>t</sup> <sup>e</sup>*�*δ<sup>s</sup>* ∥*ws*∥. If 0≤ *τ* ≤ *t*, then from Eq. (8), we have

$$\begin{split} \|e^{-\delta\tau} \|w(\tau)\|\_{a} &\leq M\_{1} e^{(\sigma-\delta)\tau} \|\varphi\_{1} - \varphi\_{2}\|\_{a} + L\_{F} N\_{a} \int\_{0}^{\tau} \frac{e^{(\sigma-\delta)(\tau-s)}}{(\tau-s)^{a}} e^{-\delta t} \|w\_{s}\|\_{a} ds \\ &\leq M\_{1} \|\varphi\_{1} - \varphi\_{2}\|\_{a} + L\_{F} M\_{1} \text{ET}(\mathbf{1}-a) (\delta-\sigma)^{a-1} .\end{split} \tag{9}$$

**130**

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

If �*r*≤ *τ* ≤ 0, we have

We have ð Þ Γ*az* ð Þ¼ *a y a*ð Þ and Γ*az* is continuous. Then it follows that Γ*a*∧⊂∧*:*

ð*t a*

1

ð*a* 0 *ds sα*

� �

*δt*

ð*t* 0

*;* <sup>1</sup> � �*LF*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup> , <sup>1</sup>*:*

�*δr ;* 1 � �*:*

*e*ð Þ *<sup>σ</sup>*�*<sup>δ</sup>* ð Þ *<sup>τ</sup>*�*<sup>s</sup>* ð Þ *<sup>τ</sup>* � *<sup>s</sup> <sup>α</sup> <sup>e</sup>*

ð*τ* 0

∥*ws*∥. If 0≤ *τ* ≤ *t*, then from Eq. (8), we have

�*δs*

*:*

∥*ws*∥*αds*

(9)

*eσ*ð Þ *<sup>t</sup>*�*<sup>s</sup>*

ð<sup>∞</sup> 0 *e* �*kss* �*αds,*

Then we deduce that Γ*<sup>a</sup>* has a unique fixed point in ∧ which extends the solution

Now we show the continuous dependence of the mild solutions with respect to

**Theorem 3.4.** Assume that (**V1**)–(**V3**) and (**H1**) hold. Then the mild solution

*U t*ð Þ*φ* ¼ *ut*ð Þ *:; φ* . That is, *U t*ð Þ*φ* is continuous from 0½ Þ ; ∞ to C*<sup>α</sup>* for each fixed *φ* ∈ C*α*. Moreover there exist a real number *δ* and a scalar function *P* such that for *t*≥0 and

*y* in ½ � *a;* 2*a* . Proceeding inductively, *y* is uniquely and continuously extended to

*u*ð Þ *:; φ* of Eq. (1) defines a continuous Lipschitz operator *U t*ð Þ*, t*≥0 in C*<sup>α</sup>* by

∥*U t*ð Þ*φ*<sup>1</sup> � *U t*ð Þ*φ*2∥ ≤ *P*ð Þ*δ e*

<sup>Γ</sup>ð Þ <sup>1</sup> � *<sup>α</sup> <sup>k</sup>α*�<sup>1</sup> <sup>¼</sup>

where *k* . 0 (see [20], p. 265). The continuity is obvious that the map *t* ! *ut*ð Þ *:; φ* is continuous. Now, let *φ*1*, φ*<sup>2</sup> ∈ C*α:* If we pose *w t*ðÞ¼ *u t; φ*<sup>1</sup> ð Þ� *u t; φ*<sup>2</sup> ð Þ*,*

∥*φ*<sup>1</sup> � *φ*2∥*<sup>α</sup>* þ *LFN<sup>α</sup>*

�*δr*

*<sup>P</sup>*ð Þ¼ *<sup>δ</sup> <sup>M</sup>*1*M*3*M*<sup>4</sup> <sup>1</sup> � *<sup>M</sup>*1*M*4*LF*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup> � ��<sup>1</sup>

*;* <sup>1</sup> � �*, M*<sup>4</sup> <sup>¼</sup> max *<sup>e</sup>*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* k k *us* � *vs <sup>α</sup> ds:*

k k *u* � *v <sup>α</sup>:*

∥*φ*<sup>1</sup> � *φ*2∥*α:* (7)

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* <sup>∥</sup>*ws*∥*<sup>α</sup> ds:* (8)

*<sup>A</sup><sup>α</sup>* k k ð Þ <sup>Γ</sup>*a*ð Þ� *<sup>u</sup>* <sup>Γ</sup>*a*ð Þ*<sup>v</sup>* ð Þ*<sup>t</sup>* <sup>≤</sup> *LFN<sup>α</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

*<sup>A</sup><sup>α</sup>* k k ð Þ <sup>Γ</sup>*a*ð Þ� *<sup>u</sup>* <sup>Γ</sup>*a*ð Þ*<sup>v</sup>* <sup>≤</sup> *LFN<sup>α</sup>*

Since *u* ¼ *v* ¼ *φ* in ½ � �*r;* 0 *,* we deduce that

½ � *na;*ð Þ *n* þ 1 *a* for all *n*≥ 1, and this ends the proof.

*Proof.* We use the gamma formula

∥*w t*ð Þ∥*<sup>α</sup>* ≤ *M*1*e*

Let *δ* a real number be such that

We define the function *P* by

Fix *<sup>t</sup>* . 0 and let *<sup>E</sup>* <sup>¼</sup> sup0 <sup>≤</sup> *<sup>s</sup>*<sup>≤</sup> *<sup>t</sup> <sup>e</sup>*�*δ<sup>s</sup>*

*<sup>e</sup>*�*δτ*∥*w*ð Þ*<sup>τ</sup>* <sup>∥</sup>*<sup>α</sup>* <sup>≤</sup> *<sup>M</sup>*1*e*ð Þ *<sup>σ</sup>*�*<sup>δ</sup> <sup>τ</sup>*

*σt*

*σ* � *δ* , 0 and *M*1max *e*

*M*<sup>3</sup> ¼ max *e*

*δr*

∥*φ*<sup>1</sup> � *φ*2∥*<sup>α</sup>* þ *LFN<sup>α</sup>*

<sup>≤</sup> *<sup>M</sup>*1∥*φ*<sup>1</sup> � *<sup>φ</sup>*2∥*<sup>α</sup>* <sup>þ</sup> *LFM*1*E*Γð Þ <sup>1</sup> � *<sup>α</sup>* ð Þ *<sup>δ</sup>* � *<sup>σ</sup> <sup>α</sup>*�<sup>1</sup>

Moreover, for *u, v* ∈ ∧, one has

the initial data.

*φ*1*, φ*<sup>2</sup> ∈ C*<sup>α</sup>* we have

then we have

where

**130**

$$\|e^{-\delta \tau}\| \|w(\tau)\|\_{a} \le \|\rho\_1 - \rho\_2\|\_{a} M\_3. \tag{10}$$

Therefore, Eqs. (9) and (10) imply that

$$\sup\_{-r \le \tau \le \frac{\pi}{4}} e^{-\delta \tau} \|w(\tau)\|\_{a} \le M\_1 M\_3 \|\varphi\_1 - \varphi\_2\|\_{\mathcal{C}\_a} + L\_F M\_1 E \Gamma(\mathbf{1} - a)(\delta - \sigma)^{a-1}.\tag{11}$$

For 0 ≤ *t*≤ *t*, we have

$$e^{-\delta t} \|w\_t\|\_a = \sup\_{-r \le \theta \le 0} e^{\delta \theta} e^{-\delta(t+\theta)} \|w(t+\theta)\|\_a$$

$$\le M\_4 \sup\_{-r \le \theta \le 0} e^{-\delta(t+\theta)} \|w(t+\theta)\|\_a \tag{12}$$

$$\le M\_4 \sup\_{-r \le \tau \le \tilde{t}} e^{-\delta \tau} \|w(\tau)\|\_a.$$

Then from Eqs. (11) and (12), we deduce that for 0 ≤ *t*≤ *t*

$$e^{-\delta t} \|w\_t\|\_a \le M\_1 M\_3 M\_4 \|\rho\_1 - \rho\_2\|\_a + L\_F M\_1 M\_3 E\Gamma(\mathbf{1} - a)(\delta - \sigma)^{a-1},$$

which implies that

$$E \le M\_1 M\_3 M\_4 \|\rho\_1 - \rho\_2\|\_a + L\_F M\_1 M\_4 E \Gamma(\mathbf{1} - a)(\delta - \sigma)^{a-1}.$$

Then the result follows.

#### **4. Local existence, blowing up phenomena, and the compactness of the flow**

We start by generalizing a result, obtained in [19] in the case of the usual norm on X ð Þ *α* ¼ 0 , in the case where *α* 6¼ 0. We take the following assumption.

(**H2**) *B t*ð Þ <sup>∈</sup> <sup>L</sup> <sup>X</sup>*β;* <sup>X</sup> � � for some 0 , *<sup>β</sup>* , 1, a.e *<sup>t</sup>*≥0 and k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *b t*ð Þk k*<sup>x</sup> <sup>β</sup>* for *x* ∈ X*β,* with *b* ∈ **L***<sup>q</sup> loc*ð Þ 0*;* ∞ where *q* . 1*=*ð Þ 1 � *β :*

**Theorem 4.1.** Assume that (**V1**)–(**V3**) and (**H2**) hold. Then for any *a* . 0*,* there exists a positive constant *M* ¼ *M a*ð Þ such that for *x* ∈ X we have

$$||A^a(R(t+h)\mathbb{1} - R(h)R(t)\mathbb{1})|| \le M \int\_0^h \frac{ds}{s^a} ||\mathbb{1}|| \text{ for } 0 \le h \le t \le a.s$$

*Proof.* Let *a* . 0 and *x* ∈ X. Then

$$\frac{d}{dt}R(t+h)\mathbf{x} = -AR(t+h)\mathbf{x} + \int\_0^{t+h} B(t+h-s)R(s)\mathbf{x} \, ds$$

$$= -AR(t+h)\mathbf{x} + \int\_0^t B(t-s)R(s+h)\mathbf{x} \, ds$$

$$+ \int\_t^{t+h} B(t+h-s)R(s)\mathbf{x} \, ds.$$

We deduce that *R t*ð Þ þ *h x* satisfies the equation of the form

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

$$\frac{d}{dt}\mathcal{y}(t) = -A\mathcal{y}(t) + \int\_0^t B(t-s)\mathcal{y}(s)ds + f(t).$$

ð Þ *Sη* ðÞ¼ *t*

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup>

*<sup>A</sup><sup>α</sup>* k k ð Þ ð Þ *<sup>S</sup><sup>η</sup>* ðÞ�*<sup>t</sup>* ð Þ *<sup>S</sup><sup>η</sup>* ð Þ *<sup>t</sup>*<sup>0</sup> <sup>≤</sup>

ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup>

ð Þ *Sη* ð Þ*t* . Now let 0 , *t*<sup>0</sup> , *t* ≤ *b* with *t*<sup>0</sup> be fixed. Then we have

ð*t*0 0

> � � � �

þ ð*t t*0

Using Theorem 4.1, it follows that

≤ *t*0*N*1*M*

þ *NαN*<sup>1</sup>

**133**

*<sup>A</sup><sup>α</sup>* k k ð Þ ð Þ *<sup>S</sup><sup>η</sup>* ð Þ�*<sup>t</sup>* ð Þ *<sup>S</sup><sup>η</sup>* ð Þ *<sup>t</sup>*<sup>0</sup>

ð*<sup>t</sup>*�*t*<sup>0</sup> 0

ð*<sup>t</sup>*�*t*<sup>0</sup> 0

lim *t*!*t* þ 0

*ds*

1 *<sup>s</sup><sup>α</sup> ds:* � � � �

As the set f g ð Þ *Sη* ð Þ *t*<sup>0</sup> : *η* ∈ *K*<sup>0</sup> is compact in Y*α*, we have

we have ð Þ *Sη* <sup>0</sup> ¼ 0 and

Then

show that *A<sup>β</sup>*

infer that *Aα*�*βA<sup>β</sup>*

ð*t* 0

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

k k ð Þ *Sη* ð Þ*t <sup>α</sup>* ≤

ð*t* 0

k k ð Þ *Sη* ð Þ*t <sup>α</sup>* ≤ *NαN*<sup>1</sup>

≤ *NαN*<sup>1</sup>

which implies that *S K*ð Þ<sup>0</sup> ⊂*K*0. We claim that fð Þ *Sη* ð Þ*t* Þ : *η* ∈ *K*0g is compact in Y*<sup>α</sup>* for fixed *t* ∈ ½ � �*r; b :* In fact, let *β* be such that 0 , *α* ≤ *β* , 1. The above estimate

compact in Y*α*. Next, we show that f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> is equicontinuous. The equicontinuity of f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> at *t* ¼ 0 follows from the above estimation of

<sup>þ</sup> *<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� � *<sup>t</sup>*<sup>0</sup> *<sup>I</sup>*

*<sup>s</sup><sup>α</sup>* <sup>þ</sup> ð Þ *R t*ð Þ� � *<sup>t</sup>*<sup>0</sup> *<sup>I</sup> <sup>A</sup><sup>α</sup>*

Notice that finding a fixed point of *S* in *K*<sup>0</sup> is equivalent to finding a mild solution of Eq. (1) in *K*0. Furthermore, *S* is a mapping from *K*<sup>0</sup> to *K*0, since if *η* ∈ *K*<sup>0</sup>

*R t*ð Þ � *s F s; η<sup>s</sup>* þ *φ<sup>s</sup>* ð Þ*ds* for 0≤ *t*≤ *b:* (13)

*<sup>A</sup><sup>α</sup>R t*ð Þ � *<sup>s</sup> F s; <sup>η</sup><sup>s</sup>* <sup>þ</sup> *<sup>φ</sup><sup>s</sup>* k k ð Þ *ds:*

ð*t* 0

ð*b* 0 *ds <sup>s</sup><sup>α</sup>* , *<sup>p</sup>*

� � is bounded in X. Since *Aα*�*<sup>β</sup>* is compact operator, we

ð*t*0 0

*<sup>A</sup>αR t*ð Þ � *<sup>s</sup> <sup>F</sup>*ð*s; <sup>η</sup><sup>s</sup>* <sup>þ</sup> *<sup>φ</sup><sup>s</sup>* k kÞ *ds:*

ð*t*0 0

k k ð Þ *Sη* ðÞ�*t* ð Þ *Sη* ð Þ *t*<sup>0</sup> *<sup>α</sup>* ¼ 0 uniformly in *η* ∈ *K*0*:*

We obtain the same results by taking *t*<sup>0</sup> be fixed with 0 , *t* , *t*<sup>0</sup> ≤ *b:* Then we claim that lim*<sup>t</sup>*!*t*<sup>0</sup> k k ð Þ *Sη* ðÞ�*t* ð Þ *Sη* ð Þ *t*<sup>0</sup> *<sup>α</sup>* ¼ 0 uniformly in *η* ∈ *K*<sup>0</sup> which means that f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> is equicontinuous. Then by Ascoli-Arzela theorem, f g *Sη* : *η* ∈ *K*<sup>0</sup>

� � is compact in <sup>X</sup>*,* hence f g ð Þ *<sup>S</sup><sup>η</sup>* ð Þ*<sup>t</sup>* : *<sup>η</sup>* <sup>∈</sup> *<sup>K</sup>*<sup>0</sup> is

*ds* ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*

*<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� � *<sup>s</sup> R t*ð Þ � *<sup>t</sup>*<sup>0</sup> *R t*ð Þ <sup>0</sup> � *<sup>s</sup> F s; <sup>η</sup><sup>s</sup>* <sup>þ</sup> *<sup>φ</sup><sup>s</sup>* k k ð Þ *ds*

*R t*ð Þ <sup>0</sup> � *s F*ð*s; η<sup>s</sup>* þ *φs*Þ*ds*

*R t*ð Þ <sup>0</sup> � *s F*ð*s; η<sup>s</sup>* þ *φs*Þ*ds*

� � � �

� � � � (14)

Then by the variation o constante formula, it follows that

$$\begin{aligned} R(t+h)\mathbf{x} &= R(t)R(h)\mathbf{x} + \int\_0^t R(t-s) \int\_s^{t+h} B(s+h-u)R(u)\mathbf{x} \, du \, ds \\ &= R(h)R(t)\mathbf{x} + \int\_0^h R(h-s) \int\_0^t B(u)R(s+t-u)\mathbf{x} \, du \, ds. \end{aligned}$$

Which yields that

$$R(t+h)\infty - R(h)R(t)\infty = \int\_0^h R(h-s)\int\_0^t B(u)R(s+t-u)\infty \,du\,ds.$$

Taking the *α*�norm, we obtain that

$$\begin{aligned} \|A^a(R(t+h)\mathbf{x} - R(h)R(t)\mathbf{x})\| &\le N\_\alpha \int\_0^h \frac{\mathbf{1}}{(h-s)^\alpha} \left\| \int\_0^t B(u)R(s+t-u)\mathbf{x} \, du \right\| \, ds \\ &\le N\_\alpha \int\_0^h \frac{\mathbf{1}}{(h-s)^\alpha} \int\_0^t b(u) \|A^\beta R(t+s-u)\mathbf{x}\| \, |du\, ds \\ &\le N\_\alpha N\_\beta \int\_0^h \frac{ds}{(h-s)^\alpha} \int\_0^t \frac{b(u)}{(t-u)^\beta} \, \|\mathbf{x}\| \, \|du\,. \end{aligned}$$

Let *p* be such that 1*=q* þ 1*=p* ¼ 1, so *p* , 1*=β:* Then it follows that

$$||A^a(R(t+h)\mathfrak{x} - R(h)R(t)\mathfrak{x})|| \le N\_a N\_\beta ||b||\_{\mathbf{L}^\mathbf{f}(\mathbf{0},a)} ||u^{-\beta}||\_{\mathbf{L}^\mathbf{f}(\mathbf{0},a)} \int\_0^h \frac{ds}{s^a} ||\mathfrak{x}||\,.$$

And the proof is complete.

The local existence result is given by the following Theorem.

**Theorem 4.2.** Suppose that (**V1**)–(**V3**), (**H0**), and (**H2**) hold. Moreover, assume that *F* defined from *J* � Ω into X is continuous where *J* � Ω is an open set in R<sup>þ</sup> � C*α*. Then for each *φ* ∈ Ω, Eq. (1) has at least one mild solution which is defined on some interval 0½ � *; b* .

*Proof.* Let *φ* ∈ Ω. For any real *ζ* ∈ *J* and *p* . 0*,* we define the following sets:

$$I\_{\zeta} = \{ t : \mathbf{0} \le t \le \zeta \} \quad \text{and} \quad H\_p = \left\{ \phi \in \mathcal{C}\_a : ||\phi||\_a \le p \right\}.$$

For *ϕ* ∈ *Hp*, we choose *ζ* and *p* such that ð Þ *t; ϕ* þ *φ* ∈ *I<sup>ζ</sup>* � *Hp* and *Hp*⊆Ω*:* By continuity of *F*, there exists *N*<sup>1</sup> ≥0 such that k k *F t*ð Þ *; ϕ* þ *φ* ≤ *N*<sup>1</sup> for ð Þ *t; ϕ* in *I<sup>ζ</sup>* � *Hp*. We consider *φ* ∈ *C*ð Þ ½ � �*r; ζ* ; Y*<sup>α</sup>* as the function defined by *φ*ðÞ¼ *t R t*ð Þ*φ*ð Þ 0 for *t* ∈ *I<sup>ζ</sup>* and *φ*<sup>0</sup> ¼ *φ*. Suppose that *p* , *p* and choose 0 , *b* , *ζ* such that

$$N\_a N\_1 \int\_0^b \frac{ds}{s^a} \le \overline{p} \quad \text{and} \quad ||\overline{\rho}\_t - \rho||\_a \le p - \overline{p} \text{ for } t \in I\_b.$$

Let *<sup>K</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>* <sup>∈</sup> *<sup>C</sup>*ð Þ ½ � �*r; <sup>b</sup>* ; <sup>Y</sup>*<sup>α</sup>* : *<sup>η</sup>*<sup>0</sup> <sup>¼</sup> 0 and k k *<sup>η</sup><sup>t</sup> <sup>α</sup>* <sup>≤</sup> *<sup>p</sup>* for <sup>0</sup> <sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup>* � �*:* Then we have *F t; φ<sup>t</sup>* þ *η<sup>t</sup>* k k ð Þ ≤ *N*<sup>1</sup> for 0 ≤ *t*≤ *b* and *η* ∈ *K*0*,* since k k *η<sup>t</sup>* þ *φ<sup>t</sup>* � *φ <sup>α</sup>* ≤ *p:* Consider the mapping *S* from *K*<sup>0</sup> to *C*ð Þ ½ � �*r; b* ; Y*<sup>α</sup>* defined by ð Þ *Sη* ð Þ¼ 0 0

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

$$(S\eta)(t) = \int\_0^t R(t-s)F(s, \eta\_s + \overline{\rho}\_s) \, ds \text{ for } 0 \le t \le b. \tag{13}$$

Notice that finding a fixed point of *S* in *K*<sup>0</sup> is equivalent to finding a mild solution of Eq. (1) in *K*0. Furthermore, *S* is a mapping from *K*<sup>0</sup> to *K*0, since if *η* ∈ *K*<sup>0</sup> we have ð Þ *Sη* <sup>0</sup> ¼ 0 and

$$\|\| (\mathcal{S}\eta)(t) \|\|\_{a} \le \int\_{0}^{t} \|\| A^{a} \mathcal{R}(t - s) F(s, \eta\_{s} + \overline{\eta}\_{s}) \|\| ds.$$

Then

*d*

*R t*ð Þ þ *h x* ¼ *R t*ð Þ*R h*ð Þ*x* þ

*R t*ð Þ þ *h x* � *R h*ð Þ*R t*ð Þ*x* ¼

Taking the *α*�norm, we obtain that

*<sup>A</sup><sup>α</sup>* k k ð Þ *R t*ð Þ <sup>þ</sup> *<sup>h</sup> <sup>x</sup>* � *R h*ð Þ*R t*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>N</sup><sup>α</sup>*

And the proof is complete.

*NαN*<sup>1</sup>

ð*b* 0 *ds*

on some interval 0½ � *; b* .

**132**

Which yields that

¼ *R h*ð Þ*R t*ð Þ*x* þ

*dt y t*ðÞ¼�*Ay t*ðÞþ

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

Then by the variation o constante formula, it follows that

ð*t* 0

ð*h* 0

> ð*h* 0

≤ *N<sup>α</sup>* ð*h* 0

≤ *NαN<sup>β</sup>*

Let *p* be such that 1*=q* þ 1*=p* ¼ 1, so *p* , 1*=β:* Then it follows that

*<sup>A</sup><sup>α</sup>* k k ð Þ *R t*ð Þ <sup>þ</sup> *<sup>h</sup> <sup>x</sup>* � *R h*ð Þ*R t*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>N</sup>αNβ*k k*<sup>b</sup>* **<sup>L</sup>***q*ð Þ <sup>0</sup>*;<sup>a</sup> <sup>u</sup>*�*<sup>β</sup>* �

The local existence result is given by the following Theorem.

and *φ*<sup>0</sup> ¼ *φ*. Suppose that *p* , *p* and choose 0 , *b* , *ζ* such that

mapping *S* from *K*<sup>0</sup> to *C*ð Þ ½ � �*r; b* ; Y*<sup>α</sup>* defined by ð Þ *Sη* ð Þ¼ 0 0

that *F* defined from *J* � Ω into X is continuous where *J* � Ω is an open set in R<sup>þ</sup> � C*α*. Then for each *φ* ∈ Ω, Eq. (1) has at least one mild solution which is defined

*Proof.* Let *φ* ∈ Ω. For any real *ζ* ∈ *J* and *p* . 0*,* we define the following sets:

*<sup>I</sup><sup>ζ</sup>* <sup>¼</sup> f g *<sup>t</sup>* : <sup>0</sup><sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>ζ</sup>* and *Hp* <sup>¼</sup> *<sup>ϕ</sup>* <sup>∈</sup> <sup>C</sup>*<sup>α</sup>* : k k*<sup>ϕ</sup> <sup>α</sup>* <sup>≤</sup> *<sup>p</sup>* � �*:*

For *ϕ* ∈ *Hp*, we choose *ζ* and *p* such that ð Þ *t; ϕ* þ *φ* ∈ *I<sup>ζ</sup>* � *Hp* and *Hp*⊆Ω*:* By continuity of *F*, there exists *N*<sup>1</sup> ≥0 such that k k *F t*ð Þ *; ϕ* þ *φ* ≤ *N*<sup>1</sup> for ð Þ *t; ϕ* in *I<sup>ζ</sup>* � *Hp*. We consider *φ* ∈ *C*ð Þ ½ � �*r; ζ* ; Y*<sup>α</sup>* as the function defined by *φ*ðÞ¼ *t R t*ð Þ*φ*ð Þ 0 for *t* ∈ *I<sup>ζ</sup>*

*<sup>s</sup><sup>α</sup>* , *<sup>p</sup>* and k k *<sup>φ</sup><sup>t</sup>* � *<sup>φ</sup> <sup>α</sup>* <sup>≤</sup> *<sup>p</sup>* � *<sup>p</sup>* for *<sup>t</sup>* <sup>∈</sup> *Ib:*

Let *<sup>K</sup>*<sup>0</sup> <sup>¼</sup> *<sup>η</sup>* <sup>∈</sup> *<sup>C</sup>*ð Þ ½ � �*r; <sup>b</sup>* ; <sup>Y</sup>*<sup>α</sup>* : *<sup>η</sup>*<sup>0</sup> <sup>¼</sup> 0 and k k *<sup>η</sup><sup>t</sup> <sup>α</sup>* <sup>≤</sup> *<sup>p</sup>* for <sup>0</sup> <sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup>* � �*:* Then we have *F t; φ<sup>t</sup>* þ *η<sup>t</sup>* k k ð Þ ≤ *N*<sup>1</sup> for 0 ≤ *t*≤ *b* and *η* ∈ *K*0*,* since k k *η<sup>t</sup>* þ *φ<sup>t</sup>* � *φ <sup>α</sup>* ≤ *p:* Consider the

ð*t* 0

*R t*ð Þ � *s*

*R h*ð Þ � *s*

*R h*ð Þ � *s*

ð*h* 0 ð*s*þ*<sup>h</sup> s*

ð*t* 0

> ð*t* 0

1 ð Þ *<sup>h</sup>* � *<sup>s</sup> <sup>α</sup>*

1 ð Þ *<sup>h</sup>* � *<sup>s</sup> <sup>α</sup>*

ð*h* 0

**Theorem 4.2.** Suppose that (**V1**)–(**V3**), (**H0**), and (**H2**) hold. Moreover, assume

ð*t* 0

� � � �

ð*t* 0

> ð*t* 0

*b u*ð Þ ð Þ *<sup>t</sup>* � *<sup>u</sup> <sup>β</sup>* <sup>∥</sup>*x*∥*du:*

� � � **<sup>L</sup>** *<sup>p</sup>*ð Þ <sup>0</sup>*;<sup>a</sup>*

*ds* ð Þ *<sup>h</sup>* � *<sup>s</sup> <sup>α</sup>*

*B t*ð Þ � *s y s*ð Þ*ds* þ *f t*ð Þ*:*

*B s*ð Þ þ *h* � *u R u*ð Þ*x du ds*

*B u*ð Þ*R s*ð Þ þ *t* � *u x du ds:*

*B u*ð Þ*R s*ð Þ þ *t* � *u x du ds:*

*B u*ð Þ*R s*ð Þ þ *t* � *u xdu*

*b u*ð Þ∥*AβR t*ð Þ <sup>þ</sup> *<sup>s</sup>* � *<sup>u</sup> <sup>x</sup>*∥*du ds*

ð*h* 0 *ds <sup>s</sup><sup>α</sup>* <sup>∥</sup>*x*∥*:*

� � � � *ds*

$$\begin{aligned} \| (S\eta)(t) \|\_{a} &\leq N\_a N\_1 \int\_0^t \frac{ds}{(t-s)^a} \\ &\leq N\_a N\_1 \int\_0^b \frac{ds}{s^a} < \overline{p} \end{aligned}$$

which implies that *S K*ð Þ<sup>0</sup> ⊂*K*0. We claim that fð Þ *Sη* ð Þ*t* Þ : *η* ∈ *K*0g is compact in Y*<sup>α</sup>* for fixed *t* ∈ ½ � �*r; b :* In fact, let *β* be such that 0 , *α* ≤ *β* , 1. The above estimate show that *A<sup>β</sup>* ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> � � is bounded in X. Since *Aα*�*<sup>β</sup>* is compact operator, we infer that *Aα*�*βA<sup>β</sup>* ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> � � is compact in <sup>X</sup>*,* hence f g ð Þ *<sup>S</sup><sup>η</sup>* ð Þ*<sup>t</sup>* : *<sup>η</sup>* <sup>∈</sup> *<sup>K</sup>*<sup>0</sup> is compact in Y*α*. Next, we show that f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> is equicontinuous. The equicontinuity of f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> at *t* ¼ 0 follows from the above estimation of ð Þ *Sη* ð Þ*t* . Now let 0 , *t*<sup>0</sup> , *t* ≤ *b* with *t*<sup>0</sup> be fixed. Then we have

$$\begin{aligned} \left\| \left\| A^a ( (S\eta)(t) - (S\eta)(t\_0) ) \right\| \right\| &\le \int\_0^{t\_0} \left\| A^a (R(t-s) - R(t-t\_0)R(t\_0-s))F(s, \eta\_t + \overline{\eta\_t}) \right\| ds \\ &+ \left\| A^a (R(t-t\_0) - I) \int\_0^{t\_0} R(t\_0-s)F(s, \eta\_s + \overline{\eta\_s}) ds \right\| \\ &+ \int\_{t\_0}^t \left\| A^a R(t-s)F(s, \eta\_t + \overline{\eta\_s}) \right\| ds. \end{aligned} \tag{14}$$

Using Theorem 4.1, it follows that

$$\begin{aligned} & \left\| A^a(\mathcal{S}\eta)(t) - (\mathcal{S}\eta)(t\_0) \right\| \\ & \le t\_0 N\_1 \mathcal{M} \int\_0^{t-t\_0} \frac{ds}{s^a} + \left\| (R(t-t\_0) - I)A^a \int\_0^{t\_0} R(t\_0-s)F(s, \eta\_s + \overline{\eta\_s})ds \right\| \\ & + N\_a N\_1 \int\_0^{t-t\_0} \frac{1}{s^a} ds. \end{aligned}$$

As the set f g ð Þ *Sη* ð Þ *t*<sup>0</sup> : *η* ∈ *K*<sup>0</sup> is compact in Y*α*, we have

$$\lim\_{t \to t\_0^+} \left\| (\mathcal{S}\eta)(t) - (\mathcal{S}\eta)(t\_0) \right\|\_a = 0 \quad \text{uniformly in } \eta \in K\_0.$$

We obtain the same results by taking *t*<sup>0</sup> be fixed with 0 , *t* , *t*<sup>0</sup> ≤ *b:* Then we claim that lim*<sup>t</sup>*!*t*<sup>0</sup> k k ð Þ *Sη* ðÞ�*t* ð Þ *Sη* ð Þ *t*<sup>0</sup> *<sup>α</sup>* ¼ 0 uniformly in *η* ∈ *K*<sup>0</sup> which means that f g ð Þ *Sη* ð Þ*t* : *η* ∈ *K*<sup>0</sup> is equicontinuous. Then by Ascoli-Arzela theorem, f g *Sη* : *η* ∈ *K*<sup>0</sup> is relatively compact in *K*0. Finally, we prove that *S* is continuous. Since *F* is continuous, given *ε* . 0, there exists *δ* . 0, such that

$$\sup\_{0 \le s \le b} \left\| \eta(s) - \hat{\eta}(s) \right\|\_{a} \le \delta \text{ implies that } \left\| F(\mathfrak{s}, \eta\_{\varepsilon} + \overline{\mathfrak{q}}\_{\varepsilon}) - F(\mathfrak{s}, \hat{\eta}(\mathfrak{s}) + \overline{\mathfrak{q}}\_{\varepsilon}) \right\| \le \varepsilon.$$

ð*t* 0

we deduce that

We claim that the set *A<sup>α</sup>* Ð*<sup>t</sup>*

ð*tnk* 0

This implies that *A<sup>α</sup>* Ð*<sup>t</sup>*

In fact, let ð Þ *tn <sup>n</sup>*≥<sup>0</sup> be a sequence of 0*; b<sup>φ</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

k k ð Þ *R t*ð Þ� þ *h* � *s R t*ð Þ � *s F s*ð Þ *; us <sup>α</sup>ds*

≤ *bφN*2*M*

*<sup>A</sup><sup>α</sup>R tnk* � *<sup>s</sup>* � �*F s*ð Þ *; us ds* !

Now using Banach-Steinhaus' theorem, we deduce that

uniformly when *h* ! 0 with respect to *t* ∈ 0*; b<sup>φ</sup>*

∥ ð*<sup>t</sup>*þ*<sup>h</sup> t*

*u t*ðÞ� *u t*ð Þ¼ <sup>0</sup> ð Þ *R t*ðÞ� *R t*ð Þ<sup>0</sup> *φ*ð Þ� 0

� ð Þ *R t*ð Þ� <sup>0</sup> � *t I*

*u*ð Þ *:; φ* is uniformly continuous. Therefore lim*<sup>t</sup>*!*b*�

nonnegative and locally integrable on 0½ Þ *; d* with

*v t*ð Þ≤

on 0ð Þ *; d :*

**135**

*a <sup>t</sup><sup>α</sup>* <sup>þ</sup> *<sup>b</sup>*

*h*≤ 0, that is, for *t* ≤ *t*0, we have

ð Þ *R h*ð Þ� *<sup>I</sup> <sup>A</sup><sup>α</sup>*

ð*h* 0 *ds*

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

*<sup>s</sup><sup>α</sup>* <sup>þ</sup> ð Þ *R h*ð Þ� *<sup>I</sup> <sup>A</sup><sup>α</sup>*

<sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; us ds* : *t* ∈ 0*, b<sup>φ</sup>*

ð*t*0 0

� � � � is relatively compact.

*R t*ð Þ � *s F s*ð Þ *; us ds* ! 0

� � � �

and a real number *t*<sup>0</sup> such that *tnk* ! *t*0. Using the dominated convergence theorem,

<sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; us ds* : *t* ∈ 0*, b<sup>φ</sup>*

*R t*ð Þ þ *h* � *s F s*ð Þ *; us ds*∥*<sup>α</sup>* ≤ *N*2*N<sup>α</sup>*

Consequently k k *u t*ð Þ� þ *h u t*ð Þ *<sup>α</sup>* ! 0 *as h* ! 0 uniformly in *t* ∈ 0*; b<sup>φ</sup>*

ð*t* 0

one can show similar results by using the same reasoning. This implies that

quently, *u*ð Þ *:; φ* can be extended to *b<sup>φ</sup>* which contradicts the maximality of 0*; b<sup>φ</sup>*

The next result gives the global existence of the mild solutions under weak conditions of *F*. To achieve our goal, we introduce a following necessary result

**Lemma 4.4.** [21] Let *α, a, b*≥0*, β* , 1 and 0 , *d* , ∞*:* Also assume that *v* is

*v s*ð Þ

Then there exists a constant *M*<sup>2</sup> ¼ *M*2ð Þ *a; b; α; β; d* , ∞ such that *v t*ð Þ≤ *M*2*=t*

**Theorem 4.5.** Assume that (**V1**)–(**V3**), (**H0**), and (**H2**) hold and *F* is a completely continuous function on R<sup>þ</sup> � C*α*. Moreover suppose that there exist continuous nonnegative functions *f* <sup>1</sup> and *f* <sup>2</sup> such that k k *F t*ð Þ *; φ* ≤ *f* <sup>1</sup>ð Þ*t* k k*φ <sup>α</sup>* þ *f* <sup>2</sup>ð Þ*t* for *φ* ∈ C*<sup>α</sup>* and *t*≥0*:* Then Eq. (1) has a mild solution which is defined for *t*≥0.

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>β</sup> ds* for *<sup>t</sup>* <sup>∈</sup> ð Þ <sup>0</sup>*; <sup>d</sup> :*

*R t*ð Þ � *s F s*ð Þ *; us ds* �

ð*t* 0

which is a consequence of Lemma 7.1.1 given in ([21], p. 197, Exo 4).

ð*t* 0

ð*t* 0 ð*t* 0

� �. Then there exist a subsequence *tnk*

*<sup>A</sup><sup>α</sup>R t*ð Þ <sup>0</sup> � *<sup>s</sup> F s*ð Þ *; us ds:*

� �. Moreover we have

ð*h* 0 *ds sα :*

ð Þ *R t*ð Þ� <sup>0</sup> � *s R t*ð Þ <sup>0</sup> � *t R t*ð Þ � *s F s*ð Þ *; us ds*

*R t*ð Þ <sup>0</sup> � *s F s*ð Þ *; us ds,*

*<sup>φ</sup> u t*ð Þ *; φ exists in* Y*α:* And conse-

ð*t*0 *t*

� � � � is relatively compact.

*R t*ð Þ � *s F*ð*s; us*Þ*ds*

� � � �*:*

� �. If

� �.

*α*

� � *k*

Then for 0 ≤ *t*≤ *b*, we have

$$\|\left( (\mathbf{S}\boldsymbol{\eta})(t) - (\mathbf{S}\boldsymbol{\hat{\eta}})(t) \right)\|\_{a} \leq N\_{a} \int\_{0}^{t} \frac{\mathbf{1}}{(t-s)^{a}} \left\| F(s, \eta\_{s} + \overline{\rho}\_{s}) - F(s, \boldsymbol{\hat{\eta}}(s) + \overline{\rho}\_{s}) \right\| ds$$

$$\leq N\_{a} \varepsilon \int\_{0}^{t} \frac{ds}{s^{a}}.$$

This yields the continuity of *S*, and using Schauder's fixed point theorem, we deduce that *S* has a fixed point. Then the proof of the theorem is complete.

The following result gives the blowing up phenomena of the mild solution in finite times.

**Theorem 4.3.** Assume that (**V1**)–(**V3**), (**H0**), and (**H2**) hold and *F* is a continuous and bounded mapping. Then for each *φ* ∈ C*α*, Eq. (1) has a mild solution *u*ð Þ *:; φ* on a maximal interval of existence �*r; b<sup>φ</sup>* � �. Moreover if *b<sup>φ</sup>* , ∞, then lim*<sup>t</sup>*!*b*� *<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* ¼ þ∞.

*Proof.* Let *u*ð Þ *:; φ* be the mild solution of Eq. (1) defined on 0½ � *; b* . Similar arguments used in the local existence result can be used for the existence of *b*<sup>1</sup> . *b* and a function *u*ð Þ *:; ub* defined from ½ � *b; b*<sup>1</sup> to Y*<sup>α</sup>* satisfying

$$u(t, u\_b(., \rho)) = R(t)u(b, \rho) + \int\_b^t R(t - s)F(s, u\_s)ds \text{ for } t \in [b, b\_1].$$

By a similar proceeding, we show that the mild solution *u*ð Þ *:; φ* can be extended to a maximal interval of existence �*r; b<sup>φ</sup>* � �. Assume that *<sup>b</sup><sup>φ</sup>* , <sup>þ</sup> <sup>∞</sup> and lim*<sup>t</sup>*!*b*� *<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* , þ ∞. There exists *N*<sup>2</sup> . 0 such that k k *F s*ð Þ *; us* ≤ *N*2*,* for *s* ∈ 0*; b<sup>φ</sup>* � �. We claim that *<sup>u</sup>*ð Þ *:; <sup>φ</sup>* is uniformly continuous. In fact, let 0 , *h*≤ *t*≤ *t* þ *h* , *bφ*. Then

$$
\mu(t+h) - \mu(t) = (R(t+h) - R(t))\rho(\mathbf{0}) + \int\_0^t (R(t+h-s) - R(t-s))F(s, u\_s) \, ds.
$$

$$
+ \int\_t^{t+h} R(t+h-s)F(s, u\_s) \, ds.
$$

By continuity of *<sup>A</sup><sup>α</sup>R t*ð Þ, we claim that *<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� <sup>þ</sup> *<sup>h</sup> R t*ð Þ *<sup>φ</sup>*ð Þ <sup>0</sup> is uniformly continuous on each compact set. Moreover, Theorem 4.1 implies that *<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� <sup>þ</sup> *<sup>h</sup>* � *<sup>s</sup> R t*ð Þ � *<sup>s</sup> F s*ð Þ! *; us* 0 uniformly in *<sup>t</sup>* when *<sup>h</sup>* ! <sup>0</sup>*:* In fact we have

$$\begin{aligned} \int\_0^t \left\|(R(t+h-s)-R(t-s))F(s,u\_s)\right\|\_\alpha ds \\ \leq \int\_0^t \left\|(R(t+h-s)-R(h)R(t-s))F(s,u\_s)\right\|\_\alpha ds \\ + \left\|(R(h)-I)A^a\int\_0^t R(t-s)F(s,u\_s)\,ds\right\|\end{aligned}$$

Then using Theorem 4.1, we obtain that

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

$$\begin{aligned} \left. \int\_0^t \left|| (R(t+h-s) - R(t-s))F(s,u\_s) \right| \right| ds \\ \leq b\_\psi N\_2 M \int\_0^h \frac{ds}{s^a} + \left\| (R(h)-I)A^a \int\_0^t R(t-s)F(s,u\_s)ds \right\|. \end{aligned}$$

We claim that the set *A<sup>α</sup>* Ð*<sup>t</sup>* <sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; us ds* : *t* ∈ 0*, b<sup>φ</sup>* � � � � is relatively compact. In fact, let ð Þ *tn <sup>n</sup>*≥<sup>0</sup> be a sequence of 0*; b<sup>φ</sup>* � �. Then there exist a subsequence *tnk* � � *k* and a real number *t*<sup>0</sup> such that *tnk* ! *t*0. Using the dominated convergence theorem, we deduce that

$$\int\_0^{t\_{nk}} A^a R(t\_{n\_k} - s) F(s, u\_s) \, ds \to \int\_0^{t\_0} A^a R(t\_0 - s) F(s, u\_s) \, ds.$$

This implies that *A<sup>α</sup>* Ð*<sup>t</sup>* <sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; us ds* : *t* ∈ 0*, b<sup>φ</sup>* � � � � is relatively compact. Now using Banach-Steinhaus' theorem, we deduce that

$$(R(h) - I)A^a \int\_0^t R(t - s)F(s, u\_s) \, ds \to \mathbf{0}$$

uniformly when *h* ! 0 with respect to *t* ∈ 0*; b<sup>φ</sup>* � �. Moreover we have

$$\|\int\_{t}^{t+h} R(t+h-s)F(s,u\_s) \, ds\|\_{a} \le N\_2 N\_a \int\_0^h \frac{ds}{s^a}.$$

Consequently k k *u t*ð Þ� þ *h u t*ð Þ *<sup>α</sup>* ! 0 *as h* ! 0 uniformly in *t* ∈ 0*; b<sup>φ</sup>* � �. If *h*≤ 0, that is, for *t* ≤ *t*0, we have

$$u(t) - u(t\_0) = (R(t) - R(t\_0))\rho(\mathbf{0}) - \int\_0^t (R(t\_0 - s) - R(t\_0 - t)R(t - s))F(s, u\_s) \, ds$$

$$- (R(t\_0 - t) - I) \int\_0^t R(t - s)F(s, u\_s) \, ds - \int\_t^{t\_0} R(t\_0 - s)F(s, u\_s) \, ds,$$

one can show similar results by using the same reasoning. This implies that *u*ð Þ *:; φ* is uniformly continuous. Therefore lim*<sup>t</sup>*!*b*� *<sup>φ</sup> u t*ð Þ *; φ exists in* Y*α:* And consequently, *u*ð Þ *:; φ* can be extended to *b<sup>φ</sup>* which contradicts the maximality of 0*; b<sup>φ</sup>* � �.

The next result gives the global existence of the mild solutions under weak conditions of *F*. To achieve our goal, we introduce a following necessary result which is a consequence of Lemma 7.1.1 given in ([21], p. 197, Exo 4).

**Lemma 4.4.** [21] Let *α, a, b*≥0*, β* , 1 and 0 , *d* , ∞*:* Also assume that *v* is nonnegative and locally integrable on 0½ Þ *; d* with

$$
\nu(t) \le \frac{a}{t^a} + b \int\_0^t \frac{\nu(s)}{\left(t - s\right)^\beta} ds \text{ for } t \in (0, d).
$$

Then there exists a constant *M*<sup>2</sup> ¼ *M*2ð Þ *a; b; α; β; d* , ∞ such that *v t*ð Þ≤ *M*2*=t α* on 0ð Þ *; d :*

**Theorem 4.5.** Assume that (**V1**)–(**V3**), (**H0**), and (**H2**) hold and *F* is a completely continuous function on R<sup>þ</sup> � C*α*. Moreover suppose that there exist continuous nonnegative functions *f* <sup>1</sup> and *f* <sup>2</sup> such that k k *F t*ð Þ *; φ* ≤ *f* <sup>1</sup>ð Þ*t* k k*φ <sup>α</sup>* þ *f* <sup>2</sup>ð Þ*t* for *φ* ∈ C*<sup>α</sup>* and *t*≥0*:* Then Eq. (1) has a mild solution which is defined for *t*≥0.

is relatively compact in *K*0. Finally, we prove that *S* is continuous. Since *F* is

ð*t* 0

> ð*t* 0 *ds sα :*

deduce that *S* has a fixed point. Then the proof of the theorem is complete.

≤ *Nαε*

1

This yields the continuity of *S*, and using Schauder's fixed point theorem, we

The following result gives the blowing up phenomena of the mild solution in

**Theorem 4.3.** Assume that (**V1**)–(**V3**), (**H0**), and (**H2**) hold and *F* is a continuous and bounded mapping. Then for each *φ* ∈ C*α*, Eq. (1) has a mild solution *u*ð Þ *:; φ*

*Proof.* Let *u*ð Þ *:; φ* be the mild solution of Eq. (1) defined on 0½ � *; b* . Similar arguments used in the local existence result can be used for the existence of *b*<sup>1</sup> . *b* and a

> ð*t b*

*<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* , þ ∞. There exists *N*<sup>2</sup> . 0 such that k k *F s*ð Þ *; us* ≤ *N*2*,* for

By continuity of *<sup>A</sup><sup>α</sup>R t*ð Þ, we claim that *<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� <sup>þ</sup> *<sup>h</sup> R t*ð Þ *<sup>φ</sup>*ð Þ <sup>0</sup> is uniformly

*<sup>A</sup><sup>α</sup>*ð Þ *R t*ð Þ� <sup>þ</sup> *<sup>h</sup>* � *<sup>s</sup> R t*ð Þ � *<sup>s</sup> F s*ð Þ! *; us* 0 uniformly in *<sup>t</sup>* when *<sup>h</sup>* ! <sup>0</sup>*:* In fact we have

� �. We claim that *<sup>u</sup>*ð Þ *:; <sup>φ</sup>* is uniformly continuous. In fact, let

*R t*ð Þ þ *h* � *s F s*ð Þ *; us ds:*

continuous on each compact set. Moreover, Theorem 4.1 implies that

<sup>þ</sup> ð Þ *R h*ð Þ� *<sup>I</sup> <sup>A</sup><sup>α</sup>* <sup>Ð</sup>*<sup>t</sup>*

k k ð Þ *R t*ð Þ� þ *h* � *s R t*ð Þ � *s F s*ð Þ *; us <sup>α</sup>ds*

≤ ð*t* 0

Then using Theorem 4.1, we obtain that

By a similar proceeding, we show that the mild solution *u*ð Þ *:; φ* can be extended

ð*t* 0

k k *η*ðÞ�*s η*^ð Þ*s <sup>α</sup>* , *δ* implies that *F s; η<sup>s</sup>* þ *φ<sup>s</sup>* ð Þ� *F s; η*^ðÞþ*s φ<sup>s</sup>* k k ð Þ , *ε:*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup> F s; <sup>η</sup><sup>s</sup>* <sup>þ</sup> *<sup>φ</sup><sup>s</sup>* ð Þ� *<sup>F</sup>*ð*s; <sup>η</sup>*^ðÞþ*<sup>s</sup> <sup>φ</sup><sup>s</sup>* k kÞ *ds*

� �. Moreover if *b<sup>φ</sup>* , ∞, then

� �. Assume that *<sup>b</sup><sup>φ</sup>* , <sup>þ</sup> <sup>∞</sup> and

k k ð Þ *R t*ð Þ� þ *h* � *s R h*ð Þ*R t*ð Þ � *s F s*ð Þ *; us <sup>α</sup>ds*

<sup>0</sup> *R t*ð Þ � *<sup>s</sup> <sup>F</sup>*ð*s; us*Þ*ds* � � �

*R t*ð Þ � *s F s*ð Þ *; us ds* for *t* ∈ ½ � *b; b*<sup>1</sup> *:*

ð Þ *R t*ð Þ� þ *h* � *s R t*ð Þ � *s F s*ð Þ *; us ds*

�

continuous, given *ε* . 0, there exists *δ* . 0, such that

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

sup 0≤ *s*≤ *b*

finite times.

lim*<sup>t</sup>*!*b*�

lim*<sup>t</sup>*!*b*�

**134**

*s* ∈ 0*; b<sup>φ</sup>*

Then for 0 ≤ *t*≤ *b*, we have

k k ð Þ *Sη* ðÞ�*t* ð Þ *Sη*^ ð Þ*t <sup>α</sup>* ≤ *N<sup>α</sup>*

on a maximal interval of existence �*r; b<sup>φ</sup>*

to a maximal interval of existence �*r; b<sup>φ</sup>*

*u t*ð Þ� þ *h u t*ðÞ¼ ð Þ *R t*ð Þ� þ *h R t*ð Þ *φ*ð Þþ 0

þ ð*<sup>t</sup>*þ*<sup>h</sup> t*

function *u*ð Þ *:; ub* defined from ½ � *b; b*<sup>1</sup> to Y*<sup>α</sup>* satisfying

*u t*ð Þ¼ *; ub*ð Þ *:; φ R t*ð Þ*u b*ð Þþ *; φ*

*<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* ¼ þ∞.

0 , *h*≤ *t*≤ *t* þ *h* , *bφ*. Then

ð*t* 0

*Proof.* Let 0*; b<sup>φ</sup>* � � be the maximal interval of existence of a mild solution *<sup>u</sup>*ð Þ *:; <sup>φ</sup>* . Assume that *b<sup>φ</sup>* , þ ∞*:* By Theorem 4.3 we have lim*t*!*t*� *<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* ¼ þ∞. Recall that the solution of Eq. (1) is given by *u*<sup>0</sup> ¼ *φ* and

For each *γ* ∈ Γ*,* we define *f <sup>γ</sup>* ∈ C*<sup>α</sup>* by *f <sup>γ</sup>* ¼ *ut :; φγ*

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

*f <sup>γ</sup>* ð Þ¼ *θ R t*ð Þ þ *θ φγ* ð Þþ 0

ð*t*þ*<sup>θ</sup>* 0

ð*<sup>t</sup>*þ*θ*�*<sup>ε</sup>* 0

*θ* ∈ ½ � �*r;* 0 *,* the set *f <sup>γ</sup>* ð Þ*θ* : *γ* ∈ Γ

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

is compact. Also we have

*μ R*ð Þ*ε*

ð*<sup>t</sup>*þ*θ*�*<sup>ε</sup>* 0

We deduce that

*t*þ*θ*�*ε*

≤

*μ*

ð*<sup>t</sup>*þ*<sup>θ</sup>*

*t*þ*θ*�*ε*

^ be fixed and *<sup>h</sup>* <sup>¼</sup> *<sup>θ</sup>* � *<sup>θ</sup>*

ð*<sup>t</sup>*þ*θ*�*<sup>ε</sup>* 0

> ð*<sup>t</sup>*þ*θ*�*<sup>ε</sup>* 0

On the other hand, for 0 , *α*≤ *β* , 1, we have

*R t*ð Þ þ *θ* � *s F*ð*s; us*ð*:; φγ* ÞÞ*ds*

*R t*ð Þ þ *θ* � *s F s; us :; φγ*

^. Then

ð*ε* 0 *ds*

≤ *N*5*M*

have

*Aα*

� �

*Aβ* ð*<sup>t</sup>*þ*<sup>θ</sup>*

Thus *A<sup>β</sup>*

family *f <sup>γ</sup>* : *γ* ∈ Γ

with *θ*

**137**

� � � � � �. We show now that for fixed

¼ 0*,*

� � � �

¼ 0*:*

� *<sup>β</sup> ds*

^ ≤ *θ* ≤ 0

*R t*ð Þ <sup>þ</sup> *<sup>θ</sup>* � *<sup>s</sup> <sup>F</sup>*ð*s; us*ð*:; φγ* ÞÞ � � �

*<sup>s</sup><sup>β</sup>* ! 0 as *<sup>ε</sup>* ! <sup>0</sup>*:*

*ds* ð Þ *<sup>t</sup>* <sup>þ</sup> *<sup>θ</sup>* � *<sup>s</sup> <sup>β</sup>*

n o is precompact in <sup>Y</sup>*α*. For any *<sup>γ</sup>* <sup>∈</sup> <sup>Γ</sup>, we have

� � � � *ds* : *γ* ∈ Γ

� � � � *ds* : *γ* ∈ Γ

k k ð Þ *R t*ð Þ� þ *θ* � *s R*ð Þ*ε R t*ð Þ þ *θ* � *ε* � *s F*ð*s; us*ð*:; φ*ÞÞ *<sup>α</sup> ds*

� � � � *ds* : *γ* ∈ Γ

*t*þ*θ*�*ε*

ð*<sup>t</sup>*þ*<sup>θ</sup>*

ð*ε* 0 *ds*

*t*þ*θ*�*ε*

≤ *NβN*<sup>5</sup>

¼ *NβN*<sup>5</sup>

� � � � *ds* : *γ* ∈ Γ � � is a bounded subset of <sup>X</sup>.

The precompactness in <sup>Y</sup>*<sup>α</sup>* now follows from the compactness of *<sup>A</sup>*�*<sup>β</sup>* : <sup>X</sup> ! <sup>Y</sup>*α*. Then the set f g ð Þ *U t*ð Þ*E* ð Þ*θ* : �*r*≤ *θ* ≤ 0 is precompacted in Y*α*. We prove that the

n o is equicontinuous. Let *<sup>γ</sup>* in <sup>Γ</sup>*,* <sup>0</sup> , *<sup>ε</sup>* , *<sup>t</sup>* � *r,* and �*r*<sup>≤</sup> *<sup>θ</sup>*

*R t*ð Þ þ *θ* � *s F s; us* ð Þ ð Þ *:; φ ds:*

ð*t*þ*<sup>θ</sup>* 0

� �

As *R t*ð Þ is compact for *t* . 0, we need only to prove that the set

*R t*ð Þ þ *θ* � *s F s; us :; φγ*

*R t*ð Þ þ *θ* � *ε* � *s F s; us :; φγ*

� � � �

*<sup>s</sup><sup>α</sup>* ! 0 as *<sup>ε</sup>* ! <sup>0</sup>*:*

*R t*ð Þ þ *θ* � *s F s; us :; φγ*

� � � �

� � � � ≤ ð*<sup>t</sup>*þ*<sup>θ</sup>*

where *μ* is the measure of non-compactness. Moreover, using Theorem 4.1, we

*R t*ð Þ� þ *θ* � *ε* � *s R*ð Þ*ε R t*ð Þ þ *θ* � *ε* � *s F*ð*s; us*ð*:; φγ* ÞÞ*ds*

� � � �

$$u(t,\boldsymbol{\rho}) = \mathcal{R}(t)\boldsymbol{\rho}(\mathbf{0}) + \int\_0^t \mathcal{R}(t-s)F(s, u\_s(.,\boldsymbol{\rho})) \, ds \text{ for } t \in \left[0, b\_{\boldsymbol{\rho}}\right).$$

Then taking the *α*�norm, we obtain

$$\|\|u(t,\boldsymbol{\varrho})\|\|\_{a} \leq \|\mathcal{R}(t)\|\|\boldsymbol{\varrho}(\mathbf{0})\|\|\_{a} + k\_{2}N\_{a} \int\_{0}^{b\_{\boldsymbol{\varrho}}} \frac{ds}{s^{a}} \, ds + k\_{1}N\_{a} \int\_{0}^{t} \frac{1}{(t-s)^{a}} \|\boldsymbol{u}\_{t}(\cdot,\boldsymbol{\varrho})\|\|\_{a} \, ds,$$

where *k*<sup>1</sup> ¼ max0<sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup><sup>φ</sup>* ∣ *f* <sup>1</sup>ð Þ*t* ∣ and *k*<sup>2</sup> ¼ max0 <sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup><sup>φ</sup>* ∣ *f* <sup>2</sup>ð Þ*t* ∣. Then we deduce that

$$\|\boldsymbol{u}(t,\boldsymbol{\varrho})\|\_{a} \leq N \|\boldsymbol{\varrho}(\mathbf{0})\|\_{a} + k\_{1}N\_{a} \int\_{0}^{t} \frac{1}{\left(t-s\right)^{a}} \sup\_{-r \leq \tau \leq s} \|\boldsymbol{u}(\tau,\boldsymbol{\varrho})\|\_{a} \, ds + k\_{2}N\_{a} \int\_{0}^{b\_{\psi}} \frac{ds}{s^{a}}.\tag{15}$$

Now we claim that the function

$$t \to \int\_0^t \frac{1}{(t-s)^a} \sup\_{-r \le \tau \le s} ||u(\tau, \rho)||\_a ds,$$

is nondecreasing. In fact, let 0≤ *t*<sup>1</sup> ≤ *t*2. Then

$$\begin{aligned} \int\_0^{t\_1} \frac{1}{\left(t\_1 - s\right)^a} \sup\_{-r \le r \le s} ||u(\tau, \rho)||\_a ds &= \int\_0^{t\_1} \frac{1}{s^a} \sup\_{-r \le r \le t\_1 - s} ||u(\tau, \rho)||\_a ds \\ &\le \int\_0^{t\_1} \frac{1}{s^a} \sup\_{-r \le r \le t\_2 - s} ||u(\tau, \rho)||\_a ds \\ &= \int\_0^{t\_2} \frac{1}{\left(t\_2 - s\right)^a} \sup\_{-r \le r \le s} ||u(\tau, \rho)||\_a ds \end{aligned}$$

which yields the result. Then it follows from Eq. (15) that

$$\sup\_{-r \le s \le t} ||u(s, \boldsymbol{\rho})||\_a \le N||\boldsymbol{\rho}(\mathbf{0})||\_a + k\_2 N\_a \int\_0^{b\_\sigma} \frac{ds}{s^a} ds + k\_1 N\_a \int\_0^t \frac{1}{\left(t - s\right)^a} \sup\_{-r \le r \le s} ||u(\boldsymbol{\tau}, \boldsymbol{\rho})||\_a ds.$$

Then using Lemma 4.4, we deduce that *u*ð Þ *:; φ* is bounded in 0*; b<sup>φ</sup>* � �. Then we obtain that lim*<sup>t</sup>*!*b*� *<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* , ∞*,* which contradicts our hypothesis. Then the mild solution is global.

We focus now to the compactness of the flow defined by the mild solutions.

**Theorem 4.6.** Assume that (**V1**)–(**V3**) and (**H0**)–(**H2**) hold. Then the flow *U t*ð Þ defined from C*<sup>α</sup>* to C*<sup>α</sup>* by *U t*ð Þ*φ* ¼ *ut*ð Þ *:; φ* is compact for *t* . *r*, where *ut*ð Þ *:; φ* denotes the mild solution starting from *φ*.

*Proof.* We use Ascoli-Arzela's theorem. Let *<sup>E</sup>* <sup>¼</sup> *φγ* : *<sup>γ</sup>* <sup>∈</sup> <sup>Γ</sup> � � be a bounded subset of C*<sup>α</sup>* and let *t* . *r* be fixed, but arbitrary. We will prove that *U t*ð Þ*E* is compact. It follows from (**H1**) and inequality Eq. (7) that there exists *N*<sup>5</sup> such that

$$\left| \left| F(t, u\_t(\boldsymbol{\rho}\_\boldsymbol{\gamma})) \right| \right| \le N\_2 \left\| u\_t(\boldsymbol{\rho}\_\boldsymbol{\gamma}) \right\| + \left\| F(t, \mathbf{0}) \right\| = N\_5 \text{ for } \boldsymbol{\gamma} \in \Gamma.$$

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

For each *γ* ∈ Γ*,* we define *f <sup>γ</sup>* ∈ C*<sup>α</sup>* by *f <sup>γ</sup>* ¼ *ut :; φγ* � �. We show now that for fixed *θ* ∈ ½ � �*r;* 0 *,* the set *f <sup>γ</sup>* ð Þ*θ* : *γ* ∈ Γ n o is precompact in <sup>Y</sup>*α*. For any *<sup>γ</sup>* <sup>∈</sup> <sup>Γ</sup>, we have

$$\int\_{\mathcal{I}} f\_{\mathcal{I}}(\theta) = R(t+\theta)\rho\_{\mathcal{I}}(\mathbf{0}) + \int\_{0}^{t+\theta} R(t+\theta-s)F(s, u\_{\mathfrak{s}}(.,\rho)) \, ds.$$

As *R t*ð Þ is compact for *t* . 0, we need only to prove that the set

$$\left\{ \int\_0^{t+\theta} R(t+\theta-s) F(s, u\_s(.,\rho\_\chi)) \, ds : \chi \in \Gamma \right\}.$$

is compact. Also we have

*Proof.* Let 0*; b<sup>φ</sup>*

Assume that *b<sup>φ</sup>* , þ ∞*:* By Theorem 4.3 we have lim*t*!*t*�

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

ð*t* 0

ð*t* 0

1 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* sup

k k *u*ð Þ *τ; φ <sup>α</sup> ds* ¼

which yields the result. Then it follows from Eq. (15) that

1 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* sup

ð*b<sup>φ</sup>* 0

*ds*

where *k*<sup>1</sup> ¼ max0<sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup><sup>φ</sup>* ∣ *f* <sup>1</sup>ð Þ*t* ∣ and *k*<sup>2</sup> ¼ max0 <sup>≤</sup> *<sup>t</sup>*<sup>≤</sup> *<sup>b</sup><sup>φ</sup>* ∣ *f* <sup>2</sup>ð Þ*t* ∣. Then we deduce that

�*r*≤ *τ* ≤ *s*

≤ ð*t*2 0 1

¼ ð*t*2 0

*ds*

We focus now to the compactness of the flow defined by the mild solutions. **Theorem 4.6.** Assume that (**V1**)–(**V3**) and (**H0**)–(**H2**) hold. Then the flow *U t*ð Þ defined from C*<sup>α</sup>* to C*<sup>α</sup>* by *U t*ð Þ*φ* ¼ *ut*ð Þ *:; φ* is compact for *t* . *r*, where *ut*ð Þ *:; φ* denotes

*Proof.* We use Ascoli-Arzela's theorem. Let *<sup>E</sup>* <sup>¼</sup> *φγ* : *<sup>γ</sup>* <sup>∈</sup> <sup>Γ</sup> � � be a bounded subset

of C*<sup>α</sup>* and let *t* . *r* be fixed, but arbitrary. We will prove that *U t*ð Þ*E* is compact. It

follows from (**H1**) and inequality Eq. (7) that there exists *N*<sup>5</sup> such that

� ≤ *N*<sup>2</sup> *ut φγ* � � � � �

*<sup>s</sup><sup>α</sup> ds* <sup>þ</sup> *<sup>k</sup>*1*N<sup>α</sup>*

*<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* , ∞*,* which contradicts our hypothesis. Then the mild

ð*<sup>b</sup><sup>φ</sup>* 0

Then using Lemma 4.4, we deduce that *u*ð Þ *:; φ* is bounded in 0*; b<sup>φ</sup>*

ð*t*1 0 1

�*r*≤ *τ* ≤ *s*

*<sup>s</sup><sup>α</sup> ds* <sup>þ</sup> *<sup>k</sup>*1*N<sup>α</sup>*

that the solution of Eq. (1) is given by *u*<sup>0</sup> ¼ *φ* and

*u t*ð Þ¼ *; φ R t*ð Þ*φ*ð Þþ 0

Then taking the *α*�norm, we obtain

k k *u t*ð Þ *; φ <sup>α</sup>* ≤ k k *R t*ð Þ k k *φ*ð Þ 0 *<sup>α</sup>* þ *k*2*N<sup>α</sup>*

k k *u t*ð Þ *; φ <sup>α</sup>* ≤ *N*k k *φ*ð Þ 0 *<sup>α</sup>* þ *k*1*N<sup>α</sup>*

1 ð Þ *<sup>t</sup>*<sup>1</sup> � *<sup>s</sup> <sup>α</sup>* sup

ð*t*1 0

sup �*r*≤ *s*≤ *t*

**136**

obtain that lim*<sup>t</sup>*!*b*�

solution is global.

Now we claim that the function

*t* ! ð*t* 0

is nondecreasing. In fact, let 0≤ *t*<sup>1</sup> ≤ *t*2. Then

�*r*≤ *τ* ≤ *s*

k k *u s*ð Þ *; φ <sup>α</sup>* ≤ *N*k k *φ*ð Þ 0 *<sup>α</sup>* þ *k*2*N<sup>α</sup>*

the mild solution starting from *φ*.

*F t; ut φγ* � � � � � � �

� � be the maximal interval of existence of a mild solution *<sup>u</sup>*ð Þ *:; <sup>φ</sup>* .

*R t*ð Þ � *s F s; us* ð Þ ð Þ *:; φ ds* for *t* ∈ 0*; b<sup>φ</sup>*

k k *u*ð Þ *τ; φ <sup>α</sup> ds,*

*<sup>s</sup><sup>α</sup>* sup �*r*<sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>*1�*<sup>s</sup>*

*<sup>s</sup><sup>α</sup>* sup �*r*<sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>*2�*<sup>s</sup>*

1 ð Þ *<sup>t</sup>*<sup>2</sup> � *<sup>s</sup> <sup>α</sup>* sup

ð*t* 0

k k *u*ð Þ *τ; φ <sup>α</sup> ds* þ *k*2*N<sup>α</sup>*

k k *u*ð Þ *τ; φ <sup>α</sup> ds*

k k *u*ð Þ *τ; φ <sup>α</sup> ds*

k k *u*ð Þ *τ; φ <sup>α</sup> ds*

�*r* ≤ *τ* ≤ *s*

k k *u*ð Þ *τ; φ <sup>α</sup> ds:*

� �. Then we

�*r*≤ *τ* ≤ *s*

1 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* sup

ð*t* 0

k þ ∥*F t*ð Þ *;* 0 ∥ ¼ *N*<sup>5</sup> for *γ* ∈ Γ*:*

1

*<sup>φ</sup>* k k *u t*ð Þ *; φ <sup>α</sup>* ¼ þ∞. Recall

� �*:*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup> us* k k ð Þ *:; <sup>φ</sup> <sup>α</sup> ds,*

ð*<sup>b</sup><sup>φ</sup>* 0

*ds <sup>s</sup><sup>α</sup> :* (15)

$$\mu\left(\left\{R(\varepsilon)\int\_{0}^{t+\theta-\varepsilon}R(t+\theta-\varepsilon-s)F(s,u\_{\varepsilon}(.,\rho\_{\gamma}))\,ds:\,\gamma\in\Gamma\right\}\right)=\mathbf{0},\mathbf{0}$$

where *μ* is the measure of non-compactness. Moreover, using Theorem 4.1, we have

$$\left\| A^a \left( \int\_0^{t+\theta-\varepsilon} R(t+\theta-\varepsilon-s) - R(\varepsilon)R(t+\theta-\varepsilon-s)F(s,u\_t(.,\rho\_\varepsilon)) \, ds \right) \right\|$$

$$\leq \int\_0^{t+\theta-\varepsilon} \left\| (R(t+\theta-s) - R(\varepsilon)R(t+\theta-\varepsilon-s))F(s,u\_t(.,\rho)) \right\|\_a \, ds$$

$$\leq N\_5 \mathcal{M} \int\_0^{\varepsilon} \frac{ds}{s^a} \to 0 \quad \text{as} \quad \varepsilon \to 0.$$

We deduce that

$$\mu\left(\left\{\int\_0^{t+\theta-\iota} R(t+\theta-s)F(s,u\_s(\cdot,\rho\_\chi))\,ds:\,\chi\in\Gamma\right\}\right) = \mathbf{0}.$$

On the other hand, for 0 , *α*≤ *β* , 1, we have

$$\begin{split} \left\| A^{\beta} \int\_{t+\theta-\varepsilon}^{t+\theta} R(t+\theta-s) F(s, u\_{\varepsilon}(., \rho\_{\mathcal{I}})) ds \right\| &\leq \int\_{t+\theta-\varepsilon}^{t+\theta} \left\| R(t+\theta-s) F(s, u\_{\varepsilon}(., \rho\_{\mathcal{I}})) \right\|\_{\rho} ds \\ &\leq N\_{\beta} N\_{5} \int\_{t+\theta-\varepsilon}^{t+\theta} \frac{ds}{(t+\theta-\varepsilon)^{\beta}} \\ &= N\_{\beta} N\_{5} \int\_{0}^{\varepsilon} \frac{ds}{s^{\beta}} \to 0 \quad \text{as} \quad \varepsilon \to 0. \end{split}$$

Thus *A<sup>β</sup>* ð*<sup>t</sup>*þ*<sup>θ</sup> t*þ*θ*�*ε R t*ð Þ þ *θ* � *s F s; us :; φγ* � � � � *ds* : *γ* ∈ Γ � � is a bounded subset of <sup>X</sup>.

The precompactness in <sup>Y</sup>*<sup>α</sup>* now follows from the compactness of *<sup>A</sup>*�*<sup>β</sup>* : <sup>X</sup> ! <sup>Y</sup>*α*. Then the set f g ð Þ *U t*ð Þ*E* ð Þ*θ* : �*r*≤ *θ* ≤ 0 is precompacted in Y*α*. We prove that the family *f <sup>γ</sup>* : *γ* ∈ Γ n o is equicontinuous. Let *<sup>γ</sup>* in <sup>Γ</sup>*,* <sup>0</sup> , *<sup>ε</sup>* , *<sup>t</sup>* � *r,* and �*r*<sup>≤</sup> *<sup>θ</sup>* ^ ≤ *θ* ≤ 0 with *θ* ^ be fixed and *<sup>h</sup>* <sup>¼</sup> *<sup>θ</sup>* � *<sup>θ</sup>* ^. Then

$$\begin{split} \left\| A^a \left( f\_r(\boldsymbol{h} + \hat{\boldsymbol{\theta}}) - f\_r(\hat{\boldsymbol{\theta}}) \right) \right\| &\leq \left\| R(\boldsymbol{t} + \hat{\boldsymbol{\theta}} + \boldsymbol{h}) - R(\boldsymbol{t} + \hat{\boldsymbol{\theta}}) \varphi\_r(\mathbf{0}) \right\|\_a \\ &\quad + \int\_0^{t + \hat{\boldsymbol{\theta}}} \left\| A^a (R(\boldsymbol{t} + \hat{\boldsymbol{\theta}} + \boldsymbol{h} - \boldsymbol{s}) - R(\boldsymbol{h}) R(\boldsymbol{t} + \hat{\boldsymbol{\theta}} - \boldsymbol{s})) F(\boldsymbol{s}, \boldsymbol{u}\_r(\boldsymbol{\cdot}, \boldsymbol{\rho}\_r)) \right\| ds \\ &\quad + \left\| (R(\boldsymbol{h}) - I) A^a \int\_0^{t + \hat{\boldsymbol{\theta}}} R(\boldsymbol{t} + \hat{\boldsymbol{\theta}} - \boldsymbol{s}) F(\boldsymbol{s}, \boldsymbol{u}\_r(\boldsymbol{\cdot}, \boldsymbol{\rho}\_r)) \right\| ds \\ &\quad + \int\_{t + \hat{\boldsymbol{\theta}}}^{t + \hat{\boldsymbol{\theta}} + \boldsymbol{h}} \left\| A^a R(\boldsymbol{t} + \hat{\boldsymbol{\theta}} + \boldsymbol{h} - \boldsymbol{s}) F(\boldsymbol{s}, \boldsymbol{u}\_t(\boldsymbol{\cdot}, \boldsymbol{\rho}\_r)) \right\| ds. \end{split}$$

*w t*ðÞ¼ *<sup>φ</sup>*ð Þþ <sup>0</sup>

*wt* ¼ *φ* þ

¼ *R t*ð Þ*F*ð Þþ 0*; φ*

*R t*ð Þ � *s F s*ð Þ *; ws ds* �

*R s*ð Þ*Aφ*ð Þ 0 *ds* ¼ *R t*ð Þ*φ*ð Þ� 0 *φ*ð Þ� 0

ð*t* 0

On the other hand, from equality Eq. (4), we have

*w t*ðÞ¼ *φ*ð Þ� 0

þ ð*t* 0 ð*s* 0

þ ð*t* 0 ð*s* 0

*w t*ðÞ¼ *R t*ð Þ*φ*ð Þþ 0

þ ð*t* 0 ð*s* 0

þ ð*t* 0 ð*s* 0

ð*t* 0

þ ð*t* 0 ð*s* 0

þ ð*t* 0 ð*s* 0

�

We deduce, for *t* ∈ ½ � 0*; a ,* that

k k *u t*ðÞ� *w t*ð Þ *<sup>α</sup>* ≤

**139**

Consequently, the maps *<sup>t</sup>* ! *wt* and *<sup>t</sup>* ! <sup>Ð</sup>*<sup>t</sup>*

*R t*ð Þ � *s F s*ð Þ *; ws ds* ¼ *R t*ð Þ*F*ð Þþ 0*; w*<sup>0</sup>

differentiable, and the following formula holds

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

Then it follows

This implies that

ð*t* 0

*R s*ð Þ*F*ð Þ 0*; φ ds* ¼

� ð*t* 0

We rewrite *w* as follows:

Then it follows that

*d dt* ð*t* 0

ð*t* 0 8 < :

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

ð*t* 0

ð*t* 0 *v s*ð Þ*ds*if *t*≥0*,*

*vs ds* for *t* ∈ ½ � 0*; a :*

ð*t* 0 ð*s* 0

ð*t* 0

*R s*ð Þ � *τ DtF*ð Þþ *τ; u<sup>τ</sup> DφF*ð Þ *τ; u<sup>τ</sup> v<sup>τ</sup>*

*R t*ð Þ � *s F s*ð Þ *; ws ds*

*DφF*ð Þ *τ; u<sup>τ</sup> v<sup>τ</sup>* � *DφF*ð Þ *τ; w<sup>τ</sup> v<sup>τ</sup>* � �*dτ ds:*

*<sup>A</sup><sup>α</sup>* k k *R t*ð Þð � *<sup>s</sup> <sup>F</sup>*ð*s; us*Þ � *<sup>F</sup>*ð*s; ws*ÞÞ *ds*

*R s*ð Þð � *τ* ½ � *DτF*ð Þ� *τ; u<sup>τ</sup> DτF*ð*τ; wτ*ÞÞ *dτ ds*

*<sup>A</sup><sup>α</sup>* k k *R s*ð Þð � *<sup>τ</sup> <sup>D</sup>τF*ð*τ; <sup>u</sup>τ*Þ � *<sup>D</sup>τF*ð*τ; <sup>w</sup>τ*ÞÞ *<sup>d</sup><sup>τ</sup> ds*

*<sup>A</sup><sup>α</sup>R s*ð Þð � *<sup>τ</sup> <sup>D</sup>φF*ð*τ; <sup>u</sup>τ*Þ � *<sup>D</sup>φF*ð*τ; <sup>w</sup>τ*ÞÞ*v<sup>τ</sup>*

� �

*R s*ð Þ*F*ð Þ 0*; φ ds*

� �*dτ ds*

ð*t* 0

ð*t* 0

ð*t* 0 ð*s* 0

*R s*ð Þ*Aφ*ð Þ 0 *ds* þ

*R s*ð Þ � *τ B*ð Þ*τ φ*ð Þ 0 *dτ ds:*

ð*t* 0 <sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; ws ds* are continuously

*R t*ð Þ � *s DtF s*ð Þþ *; ws DφF s*ð Þ *; ws vs*

*R t*ð Þ � *s DtF s*ð Þþ *; ws DφF s*ð Þ *; ws vs*

*R s*ð Þ � *τ DtF*ð Þþ *τ; w<sup>τ</sup> DφF*ð Þ *τ; w<sup>τ</sup> v<sup>τ</sup>*

*R s*ð Þ � *τ B*ð Þ*τ φ*ð Þ 0 *dτ ds:*

�*dτ ds:*

(17)

� �*ds*

� �*ds:*

� �*dτ ds:*

*φ*ð Þ*t* if � *r*≤ *t* ≤ 0*:*

Then it follows that

$$\begin{split} \left\| A^a \left( f\_{\boldsymbol{\gamma}} (\boldsymbol{h} + \hat{\boldsymbol{\theta}}) - f\_{\boldsymbol{\gamma}} (\hat{\boldsymbol{\theta}}) \right) \right\| &\leq \left\| \left( \mathcal{R} \{ \boldsymbol{t} + \hat{\boldsymbol{\theta}} + \boldsymbol{h} \} - \mathcal{R} \{ \boldsymbol{t} + \hat{\boldsymbol{\theta}} \} \right) A^a \rho\_{\boldsymbol{\gamma}} (\boldsymbol{0}) \right\| + MN\_5 (\boldsymbol{t} + \hat{\boldsymbol{\theta}}) \int\_0^h \frac{ds}{s^a} \\ &\quad + \left\| \left( \mathcal{R} (\boldsymbol{h} \boldsymbol{\}} - \boldsymbol{I} \right) A^a \int\_0^{t + \hat{\boldsymbol{\theta}}} \boldsymbol{R} \{ \boldsymbol{t} + \hat{\boldsymbol{\theta}} - \boldsymbol{s} \} \boldsymbol{F} (\boldsymbol{s}, \boldsymbol{u}\_{\boldsymbol{\epsilon}} (\boldsymbol{., \boldsymbol{\rho}\_{\boldsymbol{\gamma}}})) \, ds \right\| \\ &\quad + N\_5 N\_a \int\_0^h \frac{ds}{s^a} . \end{split}$$

Using the compactness of the set *A<sup>α</sup>* Ð*<sup>t</sup>*þ*<sup>θ</sup>* <sup>0</sup> *R t*ð Þ þ *θ* � *s F s; us :; φγ* � � � � *ds* : *γ* ∈ Γ n o and the continuity of *t* ! *R t*ð Þ*x* for *x* ∈ X, the right side of the above inequality can be made sufficiently small for *h* . 0 small enough. Then we conclude that *f <sup>γ</sup>* : *γ* ∈ Γ n o is equicontinuous. Consequently, by Ascoli-Arzela's theorem, we conclude that the set f g *U t*ð Þ*φ* : *φ* ∈ *E* is compact, which means that the operator *U t*ð Þ is compact for *t* . *r*.

#### **5. Regularity of the mild solutions**

We define the set C<sup>1</sup> *<sup>α</sup>* by C<sup>1</sup> *<sup>α</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>1</sup> ð Þ ½ � �*r;* 0 ; Y*<sup>α</sup>* as the set of continuously differentiable functions from ½ � �*r;* 0 to Y*α*. We assume the following hypothesis.

(**H3**) *F* is continuously differentiable, and the partial derivatives *DtF* and *DφF* are locally Lipschitz in the classical sense with respect to the second argument.

**Theorem 5.1.** Assume that (**V1**)–(**V3**), (**H1**), and (**H3**) hold. Let *φ* in C<sup>1</sup> *<sup>α</sup>* be such that *φ*ð Þ 0 ∈ Y and *φ*\_ð Þ¼� 0 *Aφ*ð Þþ 0 *F*ð Þ 0*; φ :* Then the corresponding mild solution *u* becomes a strict solution of Eq. (1).

*Proof.* Let *a* . 0. Take *φ* ∈ C<sup>1</sup> *<sup>α</sup>* such that *φ*ð Þ 0 ∈ Y and *φ*\_ð Þ¼� 0 *Aφ*ð Þþ 0 *F*ð Þ 0*; φ* , and let *u* be the mild solution of Eq. (1) which is defined on 0½ *,* þ ∞½. Consider the following equation:

$$\begin{cases} \boldsymbol{v}(t) = \boldsymbol{R}(t)\dot{\boldsymbol{\rho}}(\mathbf{0}) + \int\_{0}^{t} \boldsymbol{R}(t-s) \left[ D\_{t} \boldsymbol{F}(s, \boldsymbol{u}\_{s}) + D\_{\boldsymbol{\theta}} \boldsymbol{F}(s, \boldsymbol{u}\_{s}) \boldsymbol{v}\_{s} \right] ds \\ \quad + \int\_{0}^{t} \boldsymbol{R}(t-s) \boldsymbol{B}(s) \boldsymbol{\rho}(\mathbf{0}) \, ds \text{ for } t \ge \mathbf{0}, \\ \boldsymbol{v}\_{0} = \dot{\boldsymbol{\rho}}. \end{cases} \tag{16}$$

Using the strict contraction principle, we can show that there exists a unique continuous function *v* solution in 0½ � *; a* of Eq. (16). We introduce the function *w* defined by

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

$$w(t) = \begin{cases} \varrho(\mathbf{0}) + \int\_0^t v(s)ds \text{ if } \begin{aligned} t \ge \mathbf{0}, \end{aligned} \end{aligned}$$

$$\text{if } \quad -r \le t \le \mathbf{0}.$$

Then it follows

*<sup>A</sup><sup>α</sup> <sup>f</sup> <sup>γ</sup> <sup>h</sup>* <sup>þ</sup> *<sup>θ</sup>*

� �

^ � � � *<sup>f</sup> <sup>γ</sup> <sup>θ</sup>* ^ � � � � �

Then it follows that

^ � � � *<sup>f</sup> <sup>γ</sup> <sup>θ</sup>* ^ � � � � �

*<sup>A</sup><sup>α</sup> <sup>f</sup> <sup>γ</sup> <sup>h</sup>* <sup>þ</sup> *<sup>θ</sup>*

*f <sup>γ</sup>* : *γ* ∈ Γ n o

*U t*ð Þ is compact for *t* . *r*.

We define the set C<sup>1</sup>

following equation:

8 >>>>><

>>>>>:

defined by

**138**

**5. Regularity of the mild solutions**

*u* becomes a strict solution of Eq. (1). *Proof.* Let *a* . 0. Take *φ* ∈ C<sup>1</sup>

*v t*ðÞ¼ *R t*ð Þ*φ*\_ð Þþ 0

þ ð*t* 0

*v*<sup>0</sup> ¼ *φ*\_ *:*

*<sup>α</sup>* by C<sup>1</sup>

*<sup>α</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>1</sup>

ð*t* 0

*R t*ð Þ � *s B s*ð Þ*φ*ð Þ 0 *ds* for *t*≥ 0*,*

tiable functions from ½ � �*r;* 0 to Y*α*. We assume the following hypothesis.

(**H3**) *F* is continuously differentiable, and the partial derivatives *DtF* and *DφF* are locally Lipschitz in the classical sense with respect to the second argument. **Theorem 5.1.** Assume that (**V1**)–(**V3**), (**H1**), and (**H3**) hold. Let *φ* in C<sup>1</sup>

that *φ*ð Þ 0 ∈ Y and *φ*\_ð Þ¼� 0 *Aφ*ð Þþ 0 *F*ð Þ 0*; φ :* Then the corresponding mild solution

and let *u* be the mild solution of Eq. (1) which is defined on 0½ *,* þ ∞½. Consider the

Using the strict contraction principle, we can show that there exists a unique continuous function *v* solution in 0½ � *; a* of Eq. (16). We introduce the function *w*

*R t*ð Þ � *s DtF s*ð Þþ *; us DφF*ð*s; us*Þ*vs*

� �*ds*

� � � �

> þ ð*<sup>t</sup>*þ*<sup>θ</sup>* ^

þ ð*<sup>t</sup>*þ*<sup>θ</sup>* ^þ*<sup>h</sup>*

� �

Using the compactness of the set *A<sup>α</sup>* Ð*<sup>t</sup>*þ*<sup>θ</sup>*

� <sup>≤</sup> *R t* <sup>þ</sup> *<sup>θ</sup>*

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

0

*<sup>t</sup>*þ^*<sup>θ</sup>*

�<sup>≤</sup> *R t* <sup>þ</sup> *<sup>θ</sup>*

� � � � �

þ *N*5*N<sup>α</sup>*

<sup>þ</sup> ð Þ *R h*ð Þ� *<sup>I</sup> <sup>A</sup><sup>α</sup>*

ð*h* 0 *ds sα :*

be made sufficiently small for *h* . 0 small enough. Then we conclude that

� � � � �

<sup>þ</sup> ð Þ *R h*ð Þ� *<sup>I</sup> <sup>A</sup><sup>α</sup>*

^ <sup>þ</sup> *<sup>h</sup>* � � � *R t* <sup>þ</sup> *<sup>θ</sup>* ^ � �*φγ* ð Þ <sup>0</sup> � � �

> ð*<sup>t</sup>*þ*<sup>θ</sup>* ^

> > 0

^ <sup>þ</sup> *<sup>h</sup>* � � � *R t* <sup>þ</sup> *<sup>θ</sup>* ^ � � � � *<sup>A</sup>αφγ* ð Þ <sup>0</sup> � � �

> ð*<sup>t</sup>*þ*<sup>θ</sup>* ^

*R t* þ *θ*

<sup>0</sup> *R t*ð Þ þ *θ* � *s F s; us :; φγ*

n o

ð Þ ½ � �*r;* 0 ; Y*<sup>α</sup>* as the set of continuously differen-

*<sup>α</sup>* such that *φ*ð Þ 0 ∈ Y and *φ*\_ð Þ¼� 0 *Aφ*ð Þþ 0 *F*ð Þ 0*; φ* ,

0

and the continuity of *t* ! *R t*ð Þ*x* for *x* ∈ X, the right side of the above inequality can

conclude that the set f g *U t*ð Þ*φ* : *φ* ∈ *E* is compact, which means that the operator

is equicontinuous. Consequently, by Ascoli-Arzela's theorem, we

*<sup>A</sup><sup>α</sup>R t* <sup>þ</sup> *<sup>θ</sup>*

*<sup>A</sup><sup>α</sup> R t* <sup>þ</sup> *<sup>θ</sup>*

� *α*

^ � *<sup>s</sup>* � � � � *<sup>F</sup>*ð*s; us :; φγ* � � � � �

^ � *<sup>s</sup>* � �*F*ð*s; us*ð*:; φγ* ÞÞ *ds*

� � � � �

� þ *MN*<sup>5</sup> *t* þ *θ*

� � � � *ds* : *γ* ∈ Γ

^ � � <sup>ð</sup>*<sup>h</sup>*

� � � � � 0 *ds sα*

*<sup>α</sup>* be such

(16)

� *ds:*

^ � *<sup>s</sup>* � �*F*ð*s; us*ð*:; φγ* ÞÞ *ds*

�*ds*

^ <sup>þ</sup> *<sup>h</sup>* � *<sup>s</sup>* � � � *R h*ð Þ*R t* <sup>þ</sup> *<sup>θ</sup>*

*R t* þ *θ*

^ <sup>þ</sup> *<sup>h</sup>* � *<sup>s</sup>* � �*F*ð*s; us*ð*:; φγ* ÞÞ � � �

$$w\_t = \rho + \int\_0^t v\_s \, ds \text{ for } t \in [0, a].$$

Consequently, the maps *<sup>t</sup>* ! *wt* and *<sup>t</sup>* ! <sup>Ð</sup>*<sup>t</sup>* <sup>0</sup> *R t*ð Þ � *s F s*ð Þ *; ws ds* are continuously differentiable, and the following formula holds

$$\begin{aligned} \frac{d}{dt} \int\_0^t R(t-s)F(s,w\_s) \, ds &= R(t)F(\mathbf{0}, w\_0) + \int\_0^t R(t-s) \left[ D\_t F(s, w\_s) + D\_\psi F(s, w\_s) v\_s \right] ds \\ &= R(t)F(\mathbf{0}, \rho) + \int\_0^t R(t-s) \left[ D\_t F(s, w\_s) + D\_\psi F(s, w\_s) v\_s \right] ds \,. \end{aligned}$$

This implies that

$$\int\_0^t R(s)F(0,\rho)\,ds = \int\_0^t R(t-s)F(s,\nu\iota\_\iota)\,ds - \int\_0^t \int\_0^s R(s-\tau)\left[D\_tF(\tau,\nu\iota\_\tau) + D\_\eta F(\tau,\nu\iota\_\tau)\nu\_\tau\right]d\tau\,ds.$$

On the other hand, from equality Eq. (4), we have

$$-\int\_0^t R(s)A\rho(\mathbf{0})\,ds = R(t)\rho(\mathbf{0}) - \rho(\mathbf{0}) - \int\_0^t \int\_0^s R(s-\tau)B(\tau)\rho(\mathbf{0})\,d\tau ds.$$

We rewrite *w* as follows:

$$\begin{split} w(t) &= \rho(\mathbf{0}) - \int\_{0}^{t} R(s) A \rho(\mathbf{0}) \, ds + \int\_{0}^{t} R(s) F(\mathbf{0}, \rho) \, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{s} R(s - \tau) \left[ D\_{t} F(\tau, u\_{\tau}) + D\_{\phi} F(\tau, u\_{\tau}) v\_{\tau} \right] d\tau \, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{s} R(s - \tau) B(\tau) \rho(\mathbf{0}) \, d\tau \, ds. \end{split}$$

Then it follows that

$$\begin{split} w(t) &= R(t)\rho(\mathbf{0}) + \int\_{0}^{t} R(t-s)F(s,w\_{\varepsilon}) \, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{\varepsilon} R(s-\tau) [(D\_{\tau}F(\tau,u\_{\tau}) - D\_{\tau}F(\tau,w\_{\varepsilon}))] \, d\tau \, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{\varepsilon} \left( D\_{\phi}F(\tau,u\_{\tau})v\_{\varepsilon} - D\_{\phi}F(\tau,w\_{\varepsilon})v\_{\varepsilon} \right) d\tau \, ds. \end{split}$$

We deduce, for *t* ∈ ½ � 0*; a ,* that

$$\begin{split} \|\|u(t) - \boldsymbol{w}(t)\|\|\_{a} &\leq \int\_{0}^{t} \|\boldsymbol{A}^{a}\boldsymbol{R}(t-s)(\boldsymbol{F}(\boldsymbol{s},\boldsymbol{u}\_{\tau}) - \boldsymbol{F}(\boldsymbol{s},\boldsymbol{w}\_{\tau}))\|\, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{\boldsymbol{\tau}} \|\boldsymbol{A}^{a}\boldsymbol{R}(\boldsymbol{s}-\boldsymbol{\tau})(\boldsymbol{D}\_{\boldsymbol{\tau}}\boldsymbol{F}(\boldsymbol{\tau},\boldsymbol{u}\_{\tau}) - \boldsymbol{D}\_{\boldsymbol{\tau}}\boldsymbol{F}(\boldsymbol{\tau},\boldsymbol{w}\_{\tau}))\|\, d\boldsymbol{\tau} \, ds \\ &\quad + \int\_{0}^{t} \int\_{0}^{\boldsymbol{\tau}} \Big|\big|\boldsymbol{A}^{a}\boldsymbol{R}(\boldsymbol{s}-\boldsymbol{\tau})(\boldsymbol{D}\_{\boldsymbol{\theta}}\boldsymbol{F}(\boldsymbol{\tau},\boldsymbol{u}\_{\tau}) - \boldsymbol{D}\_{\boldsymbol{\theta}}\boldsymbol{F}(\boldsymbol{\tau},\boldsymbol{w}\_{\tau}))\boldsymbol{v}\_{\tau}\big|\big|d\boldsymbol{\tau} \, ds. \end{split} \tag{17}$$

The set *H* ¼ *us; ws* f g : *s* ∈ ½ � 0*; a* is compact in C*α*. Since the partial derivatives of *F* are locally Lipschitz with respect to the second argument, it is well-known that they are globally Lipschitz on *H*. Then we deduce that

where *w*<sup>0</sup> : ½ �� �*r;* 0 ½ �! 0*; π* R*, g* : R<sup>þ</sup> � R ! R and *h* : R<sup>þ</sup> ! Rþ are appropri-

� � equipped with norm j j� <sup>∞</sup> and the functions *<sup>u</sup>* and *<sup>φ</sup>* and *<sup>F</sup>*

� �*d<sup>θ</sup>* for <sup>a</sup>*:*<sup>e</sup> *<sup>x</sup>* <sup>∈</sup> ½ � <sup>0</sup>*; <sup>π</sup>* and *<sup>φ</sup>* <sup>∈</sup> <sup>C</sup>1*=*2*:*

*B t*ð Þ � *s u s*ð Þ*ds* þ *F t*ð Þ *; ut* for *t*≥0*,*

with  domain *D A*ð Þ¼ *<sup>H</sup>*<sup>2</sup>

by *u t*ðÞ¼ *w t*ð Þ *; x , φ θ*ð Þð Þ¼ *x w*0ð Þ *θ; x* for a.e *x* ∈ ½ � 0*; π* and *θ* ∈ ½ � �*r;* 0 *, t*≥0, and

� �*,*

The �*A* is a closed operator and generates an analytic compact semigroup

and k k *R*ð Þ *λ;* �*A* , *M=*j j *λ* for *λ* ∈ Λ. The operator *B t*ð Þ is closed and for *x* ∈ Y, k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *h t*ð Þk k*<sup>x</sup>* <sup>Y</sup>. The operator *<sup>A</sup>* has a discrete spectrum, the eigenvalues are *<sup>n</sup>*2,

*<sup>n</sup>*¼<sup>1</sup> *n u*h i *;en en* for *<sup>u</sup>* <sup>∈</sup> *D A*<sup>1</sup>*=*<sup>2</sup> � � <sup>¼</sup> *<sup>u</sup>* <sup>∈</sup> <sup>X</sup> :

**Lemma 6.1** [16] *Let φ* ∈ Y1*=*2*: Then φ is absolutely continuous, φ*

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>λ</sup>g*�<sup>1</sup>

∥*φ* 0 <sup>∥</sup> <sup>¼</sup> <sup>∥</sup>*A*<sup>1</sup>

<sup>2</sup>*φ*∥*:*

(**H5**) The function *g* : R<sup>þ</sup> � R ! R is continuous and Lipschitz with respect to

<sup>2</sup> <sup>þ</sup> *<sup>δ</sup>* � �∪f g<sup>0</sup> is contained in *<sup>ρ</sup>*ð Þ �*<sup>A</sup>* , the resolvent set of �*A*,

and *B t*ð Þ*x* ¼ *h t*ð Þ*Ax* ∈ X*, for* ≥0*, x* ∈ Y*:* For *α* ¼ 1*=*2, we define

*φ θ*ð Þð Þ *x*

ð Þ ½ � 0*; π* , with its usual norm

(19)

sin ð Þ *nx , n* ¼ 1*,* 2*,* ⋯.

<sup>0</sup>ð Þ 0*; π ,*

ð Þ <sup>0</sup>*; <sup>π</sup>* <sup>∩</sup> *<sup>H</sup>*<sup>1</sup>

*x*∥ for each *x* ∈ Y1*<sup>=</sup>*2. We define

ffiffi 2 *π* q

P<sup>∞</sup> *n*¼1 1 *<sup>n</sup>* h i *<sup>u</sup>;en en* <sup>∈</sup> <sup>X</sup> � �*:*

ð Þ <sup>0</sup>*;* <sup>∞</sup> and satisfies *<sup>g</sup>*1ð Þ¼ *<sup>λ</sup>* <sup>1</sup> <sup>þ</sup> *<sup>h</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* 6¼ <sup>0</sup>

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � ��<sup>1</sup> exists as a bounded operator on <sup>X</sup>,

<sup>1</sup> ð Þ*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Further, *<sup>h</sup>*<sup>∗</sup> ð Þ!*<sup>λ</sup>* 0 as

0

∈ X *and*

ate functions. To study this equation, we choose <sup>X</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

k k*:* . We define the operator *A* : Y ¼ *D A*ð Þ⊂ X ! X by

� � where j j *<sup>x</sup>* <sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>∥</sup>*A*1*=*<sup>2</sup>

*g t; <sup>∂</sup> ∂x*

0

ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on X. Thus, there exists *δ* in 0ð Þ *; π=*2 and *M* ≥ 0 such that

*<sup>u</sup>*<sup>0</sup> <sup>¼</sup> *<sup>φ</sup>* <sup>∈</sup> <sup>C</sup>1*=*<sup>2</sup> <sup>¼</sup> *<sup>C</sup>* ½ � �*r;* <sup>0</sup> *; D A*<sup>1</sup>*=*<sup>2</sup> � ��

and the corresponding normalized eigenvectors are *en*ð Þ¼ *x*

*<sup>n</sup>* h i *u;en en* for *u* ∈ X.

*<sup>n</sup>*¼<sup>1</sup> *<sup>n</sup>*<sup>2</sup>h i *<sup>u</sup>;en en <sup>u</sup>* <sup>∈</sup> *D A*ð Þ.

ð0 �*r*

Then Eq. (18) takes the abstract form

*dt* ¼ �*Au t*ðÞþ <sup>ð</sup>*<sup>t</sup>*

Moreover the following formula holds:

One also has the following result.

We assume the following assumptions. (**H4**) The scalar function *<sup>h</sup>*ð Þ*:* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup>

*<sup>h</sup>*<sup>∗</sup> <sup>ð</sup> the Laplace transform of h) and *<sup>λ</sup>g*�<sup>1</sup>

By assumption (**H4**), the operator

j j *<sup>λ</sup>* ! <sup>∞</sup>*,* for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> and *<sup>h</sup>*<sup>∗</sup> ð Þ ð Þ*<sup>λ</sup>* �<sup>1</sup> <sup>¼</sup> <sup>∘</sup> j j *<sup>λ</sup> <sup>n</sup>* ð Þ.

*<sup>u</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup> *n*¼1 1

*<sup>u</sup>* <sup>¼</sup> <sup>P</sup><sup>∞</sup>

*Aw* ¼ �*w*<sup>00</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

<sup>Y</sup>1*=*<sup>2</sup> <sup>¼</sup> *D A*1*=*<sup>2</sup> � �*;* j j� <sup>1</sup>*=*<sup>2</sup>

*F t*ð Þ *; φ* ð Þ¼ *x*

*du t*ð Þ

8 ><

>:

<sup>Λ</sup> <sup>¼</sup> *<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : j j *arg<sup>λ</sup>* , *<sup>π</sup>*

i. *Au* <sup>¼</sup> <sup>P</sup><sup>∞</sup>

the second variable.

**141**

*ρ λ*ð Þ¼ *<sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>g</sup>*1ð Þ*<sup>λ</sup> <sup>A</sup>* � ��<sup>1</sup> <sup>¼</sup> *<sup>g</sup>*�<sup>1</sup>

ii. *A*�1*=*<sup>2</sup>

iii. *A*<sup>1</sup>*=*<sup>2</sup>

C1*=*<sup>2</sup> = *C* ½ � �*r;* 0 *;* Y1*=*<sup>2</sup>

finally

$$\begin{aligned} \|u(t) - w(t)\|\_{a} &\leq N\_{a}h(a) \int\_{0}^{t} \frac{1}{(t-s)^{a}} \|u\_{s} - w\_{s}\|\_{a} \, ds \\ &\leq N\_{a}h(a) \int\_{0}^{t} \frac{1}{(t-s)^{a}} \sup\_{0 \leq \tau \leq a} \|u(\tau) - w(\tau)\|\_{a} \, ds, \end{aligned}$$

where *h a*ð Þ¼ *LFN<sup>α</sup>* <sup>þ</sup> *aNα*Lipð Þþ *DtF aNα*Lip *<sup>D</sup>φ<sup>F</sup>* � �*,* with Lip *<sup>D</sup>φ<sup>F</sup>* � � and Lipð Þ *DtF* the Lipschitz constant of *DφF* and *DtF*, respectively, which implies that

$$||u - w||\_a \le \left( N\_a h(a) \int\_0^a \frac{ds}{s^a} \right) ||u - w||\_a.$$

If we choose *a* such that

$$N\_a h(a) \int\_0^a \frac{ds}{s^a} < \mathbf{1},$$

then *u* ¼ *w* in 0½ � *; a* . Now we will prove that *u* ¼ *w* in 0½ Þ *;* þ∞ *:* Assume that there exists *t*<sup>0</sup> . 0 such that *u t*ð Þ<sup>0</sup> 6¼ *w t*ð Þ<sup>0</sup> . Let *t*<sup>1</sup> ¼ inff g *t* . 0 : ∥*u t*ðÞ� *w t*ð Þ∥ . 0 *:* By continuity, one has *u t*ðÞ¼ *w t*ð Þ for *t* ≤ *t*1, and there exists *ε* . 0 such that ∥*u t*ðÞ� *w t*ð Þ∥ . 0 for *t* ∈ ð Þ *t*1*; t*<sup>1</sup> þ *ε* . Then it follows that for *t* ∈ ð Þ *t*1*; t*<sup>1</sup> þ *ε* ,

$$||u(t) - w(t)||\_a \le N\_a h(\varepsilon) \int\_0^\varepsilon \frac{ds}{s^a} \sup\_{\tau \le \tau \le t\_1 + \varepsilon} ||u(\tau) - w(\tau)||\_a.$$

Now choosing *ε* such that

$$N\_a h(\varepsilon) \int\_0^{\varepsilon} \frac{ds}{s^a} < \mathbf{1},$$

then *u* ¼ *w* in ½ � *t*1*; t*<sup>1</sup> þ *ε* which gives a contradiction. Consequently, *u t*ðÞ¼ *w t*ð Þ for *t*≥0. We conclude that *t* ! *ut* from 0½ Þ *;* þ∞ to Y*<sup>α</sup>* and *t* ! *F t*ð Þ *; ut* from ½ Þ� 0*;* þ∞ C*<sup>α</sup>* to X are continuously differentiable. Thus, we claim that *u* is a strict solution of Eq. (1) on 0½ Þ *;* þ∞ [22–31].

#### **6. Application**

For illustration, we propose to study the model Eq. (2) given in the Introduction. We recall that this is defined by

$$\begin{cases} \frac{\partial}{\partial t} w(t, \mathbf{x}) = \frac{\partial^2}{\partial \mathbf{x}^2} w(t, \mathbf{x}) + \int\_0^t h(t - s) \frac{\partial^2}{\partial \mathbf{x}^2} w(s, \mathbf{x}) \, ds \\\\ \qquad + \int\_{-r}^0 g\left(t, \frac{\partial}{\partial \mathbf{x}} w(t + \theta, \mathbf{x})\right) d\theta \text{ for } t \ge 0 \text{ and } \mathbf{x} \in [0, \pi], \\\\ w(t, 0) = w(t, \pi) = 0 \text{ for } t \ge 0, \\\\ w(\theta, \mathbf{x}) = w\_0(\theta, \mathbf{x}) \text{ for } \theta \in [-r, 0] \text{ and } \mathbf{x} \in [0, \pi], \end{cases} \tag{18}$$

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

where *w*<sup>0</sup> : ½ �� �*r;* 0 ½ �! 0*; π* R*, g* : R<sup>þ</sup> � R ! R and *h* : R<sup>þ</sup> ! Rþ are appropriate functions. To study this equation, we choose <sup>X</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ ½ � 0*; π* , with its usual norm k k*:* . We define the operator *A* : Y ¼ *D A*ð Þ⊂ X ! X by

$$Aw = -w'\text{ with domain }D(A) = H^2(0,\pi) \cap H^1\_0(0,\pi),$$

and *B t*ð Þ*x* ¼ *h t*ð Þ*Ax* ∈ X*, for* ≥0*, x* ∈ Y*:* For *α* ¼ 1*=*2, we define <sup>Y</sup>1*=*<sup>2</sup> <sup>¼</sup> *D A*1*=*<sup>2</sup> � �*;* j j� <sup>1</sup>*=*<sup>2</sup> � � where j j *<sup>x</sup>* <sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> <sup>∥</sup>*A*1*=*<sup>2</sup> *x*∥ for each *x* ∈ Y1*<sup>=</sup>*2. We define C1*=*<sup>2</sup> = *C* ½ � �*r;* 0 *;* Y1*=*<sup>2</sup> � � equipped with norm j j� <sup>∞</sup> and the functions *<sup>u</sup>* and *<sup>φ</sup>* and *<sup>F</sup>* by *u t*ðÞ¼ *w t*ð Þ *; x , φ θ*ð Þð Þ¼ *x w*0ð Þ *θ; x* for a.e *x* ∈ ½ � 0*; π* and *θ* ∈ ½ � �*r;* 0 *, t*≥0, and finally

$$F(t, \rho)(\mathbf{x}) = \int\_{-r}^{0} \mathbf{g}\left(t, \frac{\partial}{\partial \mathbf{x}} \rho(\theta)(\mathbf{x})\right) d\theta \text{ for a.e. } \mathbf{x} \in [0, \pi] \text{ and } \rho \in \mathcal{C}\_{1/2}.$$

Then Eq. (18) takes the abstract form

The set *H* ¼ *us; ws* f g : *s* ∈ ½ � 0*; a* is compact in C*α*. Since the partial derivatives of *F* are locally Lipschitz with respect to the second argument, it is well-known that they

1

1 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* sup 0 ≤ *τ* ≤ *a*

> ð*a* 0 *ds sα*

� �

ð*a* 0 *ds <sup>s</sup><sup>α</sup>* , <sup>1</sup>*,*

then *u* ¼ *w* in 0½ � *; a* . Now we will prove that *u* ¼ *w* in 0½ Þ *;* þ∞ *:* Assume that there

*<sup>s</sup><sup>α</sup>* sup *<sup>ε</sup>*<sup>≤</sup> *<sup>τ</sup>* <sup>≤</sup> *<sup>t</sup>*1þ*<sup>ε</sup>*

ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>* k k *us* � *ws <sup>α</sup> ds*

k k *u*ð Þ� *τ w*ð Þ*τ <sup>α</sup>ds,*

k k *u*ð Þ� *τ w*ð Þ*τ <sup>α</sup>:*

k k *u* � *w <sup>α</sup>:*

ð*t* 0

ð*t* 0

where *h a*ð Þ¼ *LFN<sup>α</sup>* <sup>þ</sup> *aNα*Lipð Þþ *DtF aNα*Lip *<sup>D</sup>φ<sup>F</sup>* � �*,* with Lip *<sup>D</sup>φ<sup>F</sup>* � � and Lipð Þ *DtF* the Lipschitz constant of *DφF* and *DtF*, respectively, which implies that

≤ *Nαh a*ð Þ

k k *u* � *w <sup>α</sup>* ≤ *Nαh a*ð Þ

k k *u t*ðÞ� *w t*ð Þ *<sup>α</sup>* ≤ *Nαh*ð Þ*ε*

*Nαh a*ð Þ

exists *t*<sup>0</sup> . 0 such that *u t*ð Þ<sup>0</sup> 6¼ *w t*ð Þ<sup>0</sup> . Let *t*<sup>1</sup> ¼ inff g *t* . 0 : ∥*u t*ðÞ� *w t*ð Þ∥ . 0 *:* By continuity, one has *u t*ðÞ¼ *w t*ð Þ for *t* ≤ *t*1, and there exists *ε* . 0 such that ∥*u t*ðÞ� *w t*ð Þ∥ . 0 for *t* ∈ ð Þ *t*1*; t*<sup>1</sup> þ *ε* . Then it follows that for *t* ∈ ð Þ *t*1*; t*<sup>1</sup> þ *ε* ,

> ð*ε* 0 *ds*

> > ð*ε* 0 *ds <sup>s</sup><sup>α</sup>* , <sup>1</sup>*,*

then *u* ¼ *w* in ½ � *t*1*; t*<sup>1</sup> þ *ε* which gives a contradiction. Consequently, *u t*ðÞ¼ *w t*ð Þ

For illustration, we propose to study the model Eq. (2) given in the Introduction.

*<sup>∂</sup>x*<sup>2</sup> *w s*ð Þ *; <sup>x</sup> ds*

*dθ* for *t*≥0 and *x* ∈ ½ � 0*; π ,*

(18)

*Nαh*ð Þ*ε*

for *t*≥0. We conclude that *t* ! *ut* from 0½ Þ *;* þ∞ to Y*<sup>α</sup>* and *t* ! *F t*ð Þ *; ut* from ½ Þ� 0*;* þ∞ C*<sup>α</sup>* to X are continuously differentiable. Thus, we claim that *u* is a strict

*h t*ð Þ � *<sup>s</sup> <sup>∂</sup>*<sup>2</sup>

are globally Lipschitz on *H*. Then we deduce that

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

k k *u t*ðÞ� *w t*ð Þ *<sup>α</sup>* ≤ *Nαh a*ð Þ

If we choose *a* such that

Now choosing *ε* such that

solution of Eq. (1) on 0½ Þ *;* þ∞ [22–31].

We recall that this is defined by

*w t*ð Þ¼ *; <sup>x</sup> <sup>∂</sup>*<sup>2</sup>

þ ð0 �*r*

*<sup>∂</sup>x*<sup>2</sup> *w t*ð Þþ *; <sup>x</sup>*

*g t; <sup>∂</sup>*

*w t*ð Þ¼ *;* 0 *w t*ð Þ¼ *; π* 0 for *t*≥0*,*

ð*t* 0

*<sup>∂</sup><sup>x</sup> w t*ð Þ <sup>þ</sup> *<sup>θ</sup>; <sup>x</sup>* � �

*w*ð Þ¼ *θ; x w*0ð Þ *θ; x* for *θ* ∈ ½ � �*r;* 0 and *x* ∈ ½ � 0*; π ,*

**6. Application**

*∂ ∂t*

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

**140**

$$\begin{cases} \frac{du(t)}{dt} = -Au(t) + \int\_0^t B(t-s)u(s) \, ds + F(t, u\_t) \text{ for } t \ge 0, \\\ u\_0 = \wp \in \mathcal{C}\_{1/2} = \mathcal{C} \left( [-r, 0], D \left( A^{1/2} \right) \right), \end{cases} \tag{19}$$

The �*A* is a closed operator and generates an analytic compact semigroup ð Þ *T t*ð Þ *<sup>t</sup>*≥<sup>0</sup> on X. Thus, there exists *δ* in 0ð Þ *; π=*2 and *M* ≥ 0 such that <sup>Λ</sup> <sup>¼</sup> *<sup>λ</sup>* <sup>∈</sup> <sup>C</sup> : j j *arg<sup>λ</sup>* , *<sup>π</sup>* <sup>2</sup> <sup>þ</sup> *<sup>δ</sup>* � �∪f g<sup>0</sup> is contained in *<sup>ρ</sup>*ð Þ �*<sup>A</sup>* , the resolvent set of �*A*, and k k *R*ð Þ *λ;* �*A* , *M=*j j *λ* for *λ* ∈ Λ. The operator *B t*ð Þ is closed and for *x* ∈ Y, k k *B t*ð Þ*<sup>x</sup>* <sup>≤</sup> *h t*ð Þk k*<sup>x</sup>* <sup>Y</sup>. The operator *<sup>A</sup>* has a discrete spectrum, the eigenvalues are *<sup>n</sup>*2, and the corresponding normalized eigenvectors are *en*ð Þ¼ *x* ffiffi 2 *π* q sin ð Þ *nx , n* ¼ 1*,* 2*,* ⋯. Moreover the following formula holds:

$$\text{i.} Au = \sum\_{n=1}^{\infty} n^2 \langle u, e\_n \rangle e\_n \text{ } u \in D(A).$$

$$\text{ii.} \, A^{-1/2}u = \sum\_{n=1}^{\infty} \frac{1}{n} \langle u, e\_n \rangle e\_n \text{ for } u \in \mathbb{X}.$$

$$\text{iiii.} A^{1/2}u = \sum\_{n=1}^{\infty} n \langle u, e\_n \rangle e\_n \text{ for } u \in D\left(A^{1/2}\right) = \left\{ u \in \mathbb{X} : \sum\_{n=1}^{\infty} \frac{1}{n} \langle u, e\_n \rangle e\_n \in \mathbb{X} \right\}.$$

One also has the following result.

**Lemma 6.1** [16] *Let φ* ∈ Y1*=*2*: Then φ is absolutely continuous, φ* 0 ∈ X *and*

$$\|\boldsymbol{\varrho}^{\boldsymbol{\prime}}\| = \|\boldsymbol{A}^{\frac{1}{2}}\boldsymbol{\varrho}\|.$$

We assume the following assumptions.

(**H4**) The scalar function *<sup>h</sup>*ð Þ*:* <sup>∈</sup> *<sup>L</sup>*<sup>1</sup> ð Þ <sup>0</sup>*;* <sup>∞</sup> and satisfies *<sup>g</sup>*1ð Þ¼ *<sup>λ</sup>* <sup>1</sup> <sup>þ</sup> *<sup>h</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup>* 6¼ <sup>0</sup> *<sup>h</sup>*<sup>∗</sup> <sup>ð</sup> the Laplace transform of h) and *<sup>λ</sup>g*�<sup>1</sup> <sup>1</sup> ð Þ*<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup>. Further, *<sup>h</sup>*<sup>∗</sup> ð Þ!*<sup>λ</sup>* 0 as

j j *<sup>λ</sup>* ! <sup>∞</sup>*,* for *<sup>λ</sup>* <sup>∈</sup> <sup>Λ</sup> and *<sup>h</sup>*<sup>∗</sup> ð Þ ð Þ*<sup>λ</sup>* �<sup>1</sup> <sup>¼</sup> <sup>∘</sup> j j *<sup>λ</sup> <sup>n</sup>* ð Þ.

(**H5**) The function *g* : R<sup>þ</sup> � R ! R is continuous and Lipschitz with respect to the second variable.

By assumption (**H4**), the operator

$$\rho(\lambda) = \left(\lambda I + \mathfrak{g}\_1(\lambda)A\right)^{-1} = \mathfrak{g}\_1^{-1}(\lambda)\left(\lambda \mathfrak{g}\_1^{-1}(\lambda)I + A\right)^{-1} \text{ exists as a bounded operator on } \mathbb{X}\_+$$

which is analytic in Λ and satisfies k k *ρ λ*ð Þ , *M=*j j *λ :* On the other hand, for *x* ∈ X, we have

$$\begin{split} A\rho(\lambda)\mathbf{x} &= A\left(\lambda I + \mathbf{g}\_1(\lambda)A\right)^{-1}\mathbf{x} \\ &= \left(A + \lambda \mathbf{g}\_1^{-1}(\lambda)I - \lambda \mathbf{g}\_1^{-1}(\lambda)I\right)\left(\lambda I + \mathbf{g}\_1(\lambda)A\right)^{-1}\mathbf{x} \\ &= \mathbf{g}\_1^{-1}(\lambda)\left[I - \lambda \mathbf{g}\_1^{-1}(\lambda)\left(\lambda \mathbf{g}\_1^{-1}(\lambda)I + A\right)^{-1}\right]\mathbf{x}. \end{split}$$

Since *λg*�<sup>1</sup> <sup>1</sup> ð Þ*<sup>λ</sup> <sup>λ</sup>g*�<sup>1</sup> <sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � ��<sup>1</sup> is bounded because *<sup>g</sup>*�<sup>1</sup> <sup>1</sup> ð Þ*λ* ∈ Λ, then k k *Aρ λ*ð Þ*x* has the growth properties of *g*�<sup>1</sup> <sup>1</sup> ð Þ*λ* which tends to 1 if ∣*λ*∣ goes to infinity. Then we deduce that *Aρ λ*ð Þ ∈ Lð Þ X . Moreover, it is analytic from Λ to Lð Þ X . Now, for *x* ∈ Y, one has

$$A\rho(\lambda)\mathfrak{x} = \mathfrak{g}\_1^{-1}(\lambda)\left(\log\_1^{-1}(\lambda)I + A\right)^{-1}\mathbf{A}\mathfrak{x} \text{ and } B^\*\left(\lambda\right)\rho(\lambda)\mathfrak{x} = h^\*\left(\lambda\right)\rho(\lambda)\mathbf{A}\mathfrak{x}\dots$$

Then it follows that

$$\left| \left| A\rho(\lambda)\mathfrak{x} \right| \right| \leq \mathbf{M}/|\lambda| \left| \left| \mathfrak{x} \right| \right|\_{\mathbb{Y}} \text{ and } \left| \left| B^\* \left( \lambda \right)\rho(\lambda) \right| \right| \leq \mathbf{M}/|\lambda| \left| \left| \mathfrak{x} \right|\_{\mathbb{Y}}.$$

We deduce that *<sup>A</sup>ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> , *<sup>B</sup>*<sup>∗</sup> ð Þ¼ *<sup>λ</sup> <sup>h</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> <sup>A</sup>* <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> , and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> *:* Considering *<sup>D</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> ð Þ ½ � 0*; π* , we see that the conditions (**V1**)–(**V3**) and (**H0**) are verified. Hence the homogeneous linear equation of Eq. (18) has an analytic compact resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup>. The function *F* is continuous in the first variable from the fact that *g* is continuous in the first variable. Moreover from Lemma 6.1 and the continuity of *g*, we deduce that *F* is continuous with respect to the second argument. This yields the continuity of *F* in R<sup>þ</sup> � C1*<sup>=</sup>*2. In addition, by assumption (**H5**) we deduce that

$$\|F(t, \rho\_1) - F(t, \rho\_2)\| \le rL\_f \|\rho\_1 - \rho\_2\|\_{\mathcal{C}\_{1/2}}.$$

Then *F* is a continuous globally Lipschitz function with respect to the second argument. We obtain the following important result.

**Proposition 6.2.** Suppose that the assumptions (**H4**)–(**H5**) hold. Then Eq. (19) has a mild solution which is defined for *t*≥0.

**Author details**

Boubacar Diao<sup>1</sup>

**143**

Ayyad, Marrakesh, Moroco

\*, Khalil Ezzinbi<sup>2</sup> and Mamadou Sy<sup>1</sup>

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some…*

*DOI: http://dx.doi.org/10.5772/intechopen.88090*

\*Address all correspondence to: diaobacar@yahoo.fr

provided the original work is properly cited.

1 Laboratoire L.A.N.I, Université Gaston Berger de Saint-Louis, Saint-Louis, Senegal

2 Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

*Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

### **Author details**

which is analytic in Λ and satisfies k k *ρ λ*ð Þ , *M=*j j *λ :* On the other hand, for *x* ∈ X, we

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* � *<sup>λ</sup>g*�<sup>1</sup>

*x*

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>λ</sup>g*�<sup>1</sup> <sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � ��<sup>1</sup> h i

deduce that *Aρ λ*ð Þ ∈ Lð Þ X . Moreover, it is analytic from Λ to Lð Þ X . Now, for *x* ∈ Y,

k k *<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>≤</sup> *<sup>M</sup>=*∣*λ*∣k k*<sup>x</sup>* <sup>Y</sup> and *<sup>B</sup>*<sup>∗</sup> k k ð Þ*<sup>λ</sup> ρ λ*ð Þ <sup>≤</sup> *<sup>M</sup>=*∣*λ*∣k k*<sup>x</sup>* <sup>Y</sup>*:*

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* � � *<sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>g</sup>*1ð Þ*<sup>λ</sup> <sup>A</sup>* � ��<sup>1</sup>

<sup>1</sup> ð Þ*λ* which tends to 1 if ∣*λ*∣ goes to infinity. Then we

*Ax* and *<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ*<sup>x</sup>* <sup>¼</sup> *<sup>h</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ*Ax:*

<sup>0</sup> ð Þ ½ � 0*; π* , we see that the conditions

*:*

*x*

<sup>1</sup> ð Þ*λ* ∈ Λ, then k k *Aρ λ*ð Þ*x* has

*x:*

*<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>¼</sup> *<sup>A</sup> <sup>λ</sup><sup>I</sup>* <sup>þ</sup> *<sup>g</sup>*1ð Þ*<sup>λ</sup> <sup>A</sup>* � ��<sup>1</sup>

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

<sup>¼</sup> *<sup>A</sup>* <sup>þ</sup> *<sup>λ</sup>g*�<sup>1</sup>

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* � *<sup>λ</sup>g*�<sup>1</sup>

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � ��<sup>1</sup>

R<sup>þ</sup> � C1*<sup>=</sup>*2. In addition, by assumption (**H5**) we deduce that

argument. We obtain the following important result.

has a mild solution which is defined for *t*≥0.

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>I</sup>* <sup>þ</sup> *<sup>A</sup>* � ��<sup>1</sup> is bounded because *<sup>g</sup>*�<sup>1</sup>

We deduce that *<sup>A</sup>ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> , *<sup>B</sup>*<sup>∗</sup> ð Þ¼ *<sup>λ</sup> <sup>h</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> <sup>A</sup>* <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> , and

(**V1**)–(**V3**) and (**H0**) are verified. Hence the homogeneous linear equation of Eq. (18) has an analytic compact resolvent operator ð Þ *R t*ð Þ *<sup>t</sup>*≥<sup>0</sup>. The function *F* is continuous in the first variable from the fact that *g* is continuous in the first variable. Moreover from Lemma 6.1 and the continuity of *g*, we deduce that *F* is continuous with respect to the second argument. This yields the continuity of *F* in

∥*F t; φ*<sup>1</sup> ð Þ� *F t; φ*<sup>2</sup> ð Þ∥ ≤ *rLf* ∥*φ*<sup>1</sup> � *φ*2∥C1*=*<sup>2</sup>

Then *F* is a continuous globally Lipschitz function with respect to the second

**Proposition 6.2.** Suppose that the assumptions (**H4**)–(**H5**) hold. Then Eq. (19)

<sup>¼</sup> *<sup>g</sup>*�<sup>1</sup>

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>λ</sup>g*�<sup>1</sup>

*<sup>B</sup>*<sup>∗</sup> ð Þ*<sup>λ</sup> ρ λ*ð Þ <sup>∈</sup> <sup>L</sup>ð Þ <sup>Y</sup>*;* <sup>X</sup> *:* Considering *<sup>D</sup>* <sup>¼</sup> *<sup>C</sup>*<sup>∞</sup>

have

Since *λg*�<sup>1</sup>

one has

**142**

<sup>1</sup> ð Þ*<sup>λ</sup> <sup>λ</sup>g*�<sup>1</sup>

the growth properties of *g*�<sup>1</sup>

*<sup>A</sup>ρ λ*ð Þ*<sup>x</sup>* <sup>¼</sup> *<sup>g</sup>*�<sup>1</sup>

Then it follows that

Boubacar Diao<sup>1</sup> \*, Khalil Ezzinbi<sup>2</sup> and Mamadou Sy<sup>1</sup>

1 Laboratoire L.A.N.I, Université Gaston Berger de Saint-Louis, Saint-Louis, Senegal

2 Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, Marrakesh, Moroco

\*Address all correspondence to: diaobacar@yahoo.fr

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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[16] Travis CC, Webb GF. Existence, stability and compactness in the *α*-norm for partial functional differential equations. Transactions of the American Mathematical Society. 1978;**240**:129-143

[17] A. Pazy, Semigroups of linear operators and application to partial differential equations, Applied Mathematical Sciences Vol. 44, Springer-Verlag, New York, (2001). *Existence, Regularity, and Compactness Properties in the* α*-Norm for Some… DOI: http://dx.doi.org/10.5772/intechopen.88090*

[18] Grimmer R. Resolvent operators for integral equations in a Banach space. Transactions of the American Mathematical Society. 1982;**273**:333-349

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234-259

[1] Grimmer R, Pritchard AJ. Analytic resolvent operators for integral

*Nonlinear Systems - Theoretical Aspects and Recent Applications*

functional integrodifferential equations. Afr. Diaspora J. Math. 2011;**12**(1):34-45

[10] Hannsgen KB. The resolvent kernel of an integrodifferential equation in Hilbert space. SIAM Journal on Mathematical Analysis. 1976;**7**(4):

[11] Smart DR. Uniform *L*<sup>1</sup> behavior for an integrodifferential equation with

Mathematical Analysis. 1977;**8**:626-639

[13] Miller RK. An integrodifferential equation for rigid heat conductors with memory. Journal of Mathematical Analysis and Applications. 1978;**66**:

[14] Miller RK, Wheeler RL. Asymptotic behavior for a linear Volterra integral equation in Hilbert space. Journal of Differential Equations. 1977;**23**:270-284

[15] Miller RK. Well-posedness and

integrodifferential equations in abstract spaces. Fako de l'Funkcialaj Ekvacioj Japana Matematika Societo. 1978;**21**:

[16] Travis CC, Webb GF. Existence, stability and compactness in the *α*-norm

equations. Transactions of the American Mathematical Society. 1978;**240**:129-143

for partial functional differential

[17] A. Pazy, Semigroups of linear operators and application to partial differential equations, Applied Mathematical Sciences Vol. 44, Springer-Verlag, New York, (2001).

stability of linear Volterra

parameter. SIAM Journal on

[12] Miller RK. Volterra integral equations in a Banach space. Fako de l'Funkcialaj Ekvacioj Japana Matematika

Societo. 1975;**18**:2163-2193

481-490

313-332

279-305

equations in a Banach space. Journal of Differential Equations. 1983;**50**(2):

[2] Chen G, Grimmer R. Semigroup and integral equations. Journal of Integral

[3] Hale JK, Lunel V. Introduction to functional differential equations. In: Applied Mathematical Sciences. Vol. 99. New York: Springer-Verlag; 1993

[4] Pruss J. Evolutionary Integral Equations and Applications. In: Lecture Notes in Mathematics. New York: Springer, Birkhauser; 1993

stability for partial functional

1974;**200**:395-418

[5] Travis CC, Webb GF. Existence and

differential equations. Transactions of the American Mathematical Society.

[6] Wu J. Theory and applications of partial functional differential equations. In: Applied Mathematical Sciences. Vol. 119. New York: Springer-Verlag; 1996

[7] Ezzinbi K, Touré H, Zabsonre I. Existence and regularity of solutions for

[8] Ezzinbi K, Touré H, Zabsonre I. Local existence and regularity of solutions for some partial functional integrodifferential equations with infinite delay in Banach spaces. Nonlinear Analysis. 2009;**70**(9):

[9] Ezzinbi K, Ghnimi S. Local existence and global continuation for some partial

integrodifferential equations in Banach spaces. Nonlinear Analysis: Theory Methods & Applications. 2009;**70**(7):

some partial functional

2761-2771

3378-3389

**144**

Equations. 1980;**2**(2):133-154

[19] Desch W, Grimmer R, Schappacher W. Some considerations for linear integrodifferential equations. Journal of Mathematical Analysis and Applications. 1984;**104**:219-234

[20] Hale JK. Functional Differential Equation. New York: Springer-Verlag; 1971

[21] Henry D. Geometric Theory of Semilinear Parabolic Equations. Berlin/ Heidelberg/New York: Springer-Verlag; 1981

[22] Adimy M, Ezzinbi K. Existence and linearized stability for partial neutral functional differential equations. Differential Equations and Dynamical Systems. 1999;**7**(4):371-417

[23] Engel KJ, Nagel R. One parameter semigroups of linear evolution equations. In: Brendle S, Campiti M, Hahn T, Metafune G, Nickel G, Pallara D, Perazzoli C, Rhandi A, Romanelli S, Schnaubelt R, editors. Graduate Texts in Mathematics. Vol. 194. New York: Springer-Verlag, Elsevier; 2001

[24] Ezzinbi K, Benkhalti R. Existence and stability in the *α* norm for some partial functional differential equations with infinite delay. Differential and Integral Equations. 2006;**19**(5):545-572

[25] Ezzinbi K, N'Guérékata GM. Almost automorphic solutions for some partial functional differential equations. Journal of Mathematical Analysis and Applications. 2007;**328**:344-358

[26] Goldstein JA. Some remarks on infinitesimal generators of analytic semigroups. American Mathematical Society. 1969;**22**(1):91-93

[27] Grimmer R, Pruss R. On linear Voltera equations in Banach spaces, hyperbolic partial differential equations, II. Computers and Mathematics with Applications; **11**(1–3):189-205

[28] Haraux A, Cazenave T. An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications. Vol. 131998

[29] Lunardi A. Analytic Semigroup and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications. Vol. 16. Basel: Birkhäuser Verlag; 1995

[30] Smart DR. Fixed Point Theorems, Cambridge Tracts in Mathematics. Vol. 66. London/New York: Cambridge University Press; 1974

[31] Travis CC, Webb GF. Partial differential equations with deviating arguments in the time variable. Journal of Mathematical Analysis and Applications. 1976;**56**:397-409

Section 2

Recent Applications

in Nonlinear Systems

**147**

Section 2
