Preface

In mathematics and science, a nonlinear system is one in which the change of the output is not proportional to the change of input. Because most systems in nature are nonlinear, nonlinear problems have aroused the interest of engineers, biologists, physicists, mathematicians, and many other scientists. The most prominent difference between a nonlinear system and a linear system is that a nonlinear system may lead to chaos, unpredictability, or non-intuitive results.

In general, the behavior of a nonlinear system can be mathematically described as a set of nonlinear simultaneous equations, in which the unknown number (or the unknown functions in the differential equations) appears as a polynomial variable higher than the first degree, or as a parameter of a polynomial function of a non-first degree. Generally speaking, the behavior of nonlinear systems is described mathematically by a set of nonlinear simultaneous equations, which contain non-first-degree polynomials composed of unknowns. In other words, a nonlinear equation cannot be written as a linear combination of its unknowns. A nonlinear differential equation refers to a term in which the power of the unknown function and its derivative function is not equal to one. When determining whether an equation is linear or nonlinear, only the part of the unknown number (or unknown function) needs to be considered, without checking whether there is a known nonlinear term in the equation. For example, in the differential equation, if the order of all unknown functions and unknown derivatives is one, it can still be regarded as a linear differential equation even if there is a nonlinear function composed of a known variable.

Since nonlinear equations are quite difficult to solve, we often need to approximate a nonlinear system with linear equations (linear approximation). This approximation is very accurate for the input values (variables) in a certain range, but after linear approximation, many interesting phenomena, such as solitary waves, chaos, and singularities cannot be explained. These strange phenomena often make the behavior of nonlinear systems seem counterintuitive, unpredictable, or even chaotic. Therefore, it is of great theoretical significance to analyze and solve nonlinear differential equations.

At the same time, economic advances and the continuous innovation and development of electronic computer technology have led to the rise of automatic control technology which is gradually being applied to aviation detection, engineering production, mechanical equipment research and development and management, weapons manufacturing and other areas, where it is of inestimable value. Automatic control technology has also begun to expand into the realm of social life, in areas such as biological manufacturing, medical research, and environmental management. Automatic control systems may be linear or nonlinear. A linear system has regularity, but it is nonlinear systems that are widely found in society. Compared with a linear system, a nonlinear system cannot meet the superposition, and it is unbalanced. Research into nonlinear control systems and related control strategies is of important practical significance and value for engineering applications.

This book investigates promising and in-depth research on nonlinear systems and related control strategies. Chapter 1 describes the extension of differential equations to different underlying time domains, so-called dynamic equations on time scales, whose calculus unifies the continuous and discrete calculus and extends it to any non-empty closed subset of real numbers. Dynamic equations on time scales allow the modeling of processes that are neither fully discrete nor fully continuous. Chapter 2 deals with a class of nonlinear fractional differential equations involving the Caputo fractional derivative with nonlocal boundary conditions. By applying the Leray‒Schauder nonlinear alternative and Banach contraction principle, the existence and uniqueness of this problem are established. In Chapter 3, the dynamical behavior of the incommensurate fractional-order FitzHugh‒Nagumo model of neurons is explored in detail from local stability analysis. The FitzHugh‒Nagumo model is a mathematical simplification of the Hodgkin‒Huxley model, which proves that the fractional-order FitzHugh‒Nagumo model can be simulated by a simple electrical circuit where the capacitor and the inductor are replaced by corresponding fractional-order electrical elements. The local stability of this model is then studied using the theorem on the stability of an incommensurate fractional-order system combined with Cauchy's argument principle. Finally, the dynamic behaviors of the model are investigated. In Chapter 4, an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. Interior and up-to-boundary Schauder-type estimates are established with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem is proved, and the properties of the Riesz potential regarding these Sobolev spaces are studied. In Chapter 5, a practical introduction to a generalized nonlinear analysis framework tailored for time-series data is provided, enabling the safe quantification of underlying evolutionary dynamics, which describe the referring empirical data-generating process. The reader can incorporate the proposed analysis framework, conduct the analyses and reconstructions using the correct specifications, and learn about misleading propositions or parameter choices. Chapter 6 proposes random matrix theory (RMT) to handle this problem, which begins by modeling spatial-temporal datasets as sequences, each of whose terms is in the form of a random matrix. Fundamental RMT principles are briefly discussed, such as asymptotic spectrum laws, transforms, convergence rate, and free probability, in order to extract high-dimensional statistics from the random matrix as the indicators. The statistical properties of these indicators are discussed for a better understanding of the system, and potential applications are suggested. Chapter 7 studies the structural properties and convergence approach of chance-constrained optimization of boundary-value elliptic partial differential equation systems (CCPDEs). Real-world systems, such as physical and living systems, are generally subject to vibrations that can affect their long-term integrity and safety. Thus, the determination of the law that governs the evolution of the oscillatory quantity has become a major topic in modern engineering design. Chapter 8 reviews recent developments and advances in the theory of isochronous oscillations of nonlinear systems. Chapter 9 focuses on conventional and proposed control configuration selection methods for nonlinear systems. The proposed input-output pair method for nonlinear benchmark processes is also calculated. In Chapter 10, feedback linearization techniques, including input-state and input-output linearization methods, are described. The input-output linearization method is then used for the output voltage control of an interleaved boost converter.

**V**

Chapter 11 proposes an advanced nonlinear model predictive control solution, including multi-step prediction models, for the yaw control system of a horizontal-axis wind turbine. A notable feature of the proposed solution is the use of a finite control set under constraints for the potentially demanded yaw rate; the optimal control demand for the yaw system is conveniently solved using an exhaustive search method based

In summary, this book aims to provide advanced research on nonlinear systems and

**Bo Yang**

Kunming, China

**Dušan Stipanović**

Urbana, Illinois, USA

Faculty of Electric Power Engineering,

Kunming University of Science and Technology,

University of Illinois at Urbana-Champaign,

Coordinated Science Laboratory and ISE Department,

control schemes for researchers and engineers working in relevant fields.

on a sequential diagram.

Chapter 11 proposes an advanced nonlinear model predictive control solution, including multi-step prediction models, for the yaw control system of a horizontal-axis wind turbine. A notable feature of the proposed solution is the use of a finite control set under constraints for the potentially demanded yaw rate; the optimal control demand for the yaw system is conveniently solved using an exhaustive search method based on a sequential diagram.

In summary, this book aims to provide advanced research on nonlinear systems and control schemes for researchers and engineers working in relevant fields.

#### **Bo Yang**

Faculty of Electric Power Engineering, Kunming University of Science and Technology, Kunming, China

#### **Dušan Stipanović**

Coordinated Science Laboratory and ISE Department, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Section 1

Analysis and Solution of

Nonlinear Differential

Equations

**1**

### Section 1

## Analysis and Solution of Nonlinear Differential Equations

#### **Chapter 1**

## Dynamic Equations on Time Scales

*Sabrina Streipert*

#### **Abstract**

An extension of differential equations to different underlying time domains are so called dynamic equations on time scales. Time scales calculus unifies the continuous and discrete calculus and extends it to any nonempty closed subset of the real numbers. Choosing the time scale to be the real numbers, a dynamic equation on time scales collapses to a differential equation, while the integer time scale transforms a dynamic equation into a difference equation. Dynamic equations on time scales allow the modeling of processes that are neither fully discrete nor fully continuous. This chapter provides a brief introduction to time scales and its applications by incorporating a selective collection of existing results.

**Keywords:** time scales, existence, uniqueness, linear, applications

#### **1. Introduction**

The modeling of processes using differential equations is a well-established method in multiple branches of sciences. Dependent on the model assumptions, the form of the differential equation can range from a comparably simple ordinary differential equation to more advanced formulations using nonlinear, higher order, and partial differential equations. Reasons to consider difference equations include computational benefits and, even more fundamental, a discrete modeling perspective. For example, when describing a zero-coupon bond where the invested amount at time *t*, *Mt*, receives interest *r* at the end of each year but remains unchanged during each year, the recursive model *Mt*þ<sup>1</sup> ¼ ð Þ 1 þ *r Mt* captures the change of the investment from time *t* to time *t* þ 1. Difference equations are also a common tool to describe processes on a macro scale in time, for example, when describing non-overlapping generations. Even though the number of individuals may vary throughout the generation period, one may only be interested in the individuals at the beginning of each generation time, i.e., the size of each cohort. There are however processes that cannot be described accurately using differential or difference equations. For example, when modeling species that are reproducing continuously during certain months of the year before laying eggs right before hibernating. Another example are plant populations that grow continuously during some months of the year and plant their seeds prior to dying out. In [1], Robert May gives examples of insects that exhibit such hybrid continuous–discrete behavior.

Instead of introducing a set of simplifying assumptions and possibly discontinuous model parameters that impact the model analysis, dynamic equations on time scales can provide a simple alternative to describe such processes. Time scales calculus was introduced by Stefan Hilger in 1988 [2]. It unifies the continuous and discrete calculus and extends it to any nonempty closed subset of the real numbers called a time scale, denoted by . By introducing differentiation and integration on , the classical theory of differential equations can be extended to time scales, which allows the modeling of processes that are not changing continuously nor solely discretely in time. These so-called dynamic equations are essentially the time scales analogue of differential and difference equations and have gained increasing interest due to their potential in applications. Choosing the time scale to be the real numbers, a dynamic equation transforms into a differential equation and by choosing the time scale to be the integers, a corresponding difference equation is obtained. Thus, instead of studying differential equations and difference equations separately, time scales provides also a tool to investigate both by analyzing the corresponding dynamic equation. This is specifically interesting since certain difference equations exhibit significantly different behavior as their continuous analogues, see for example the "logistic map" and the "logistic differential equation". By analyzing a dynamic equation on time scales, the effect of the underlying time domain onto the behavior of solutions may be revealed.

#### **2. Time scales fundamentals**

In this subsection, the basic definitions of time scales calculus are introduced based on the introductory book [3].

**Definition 1.** A time scale, denoted by , is a nonempty closed subset of .

Examples of a time scale are ,, *<sup>h</sup>*, *<sup>q</sup>*<sup>0</sup> <sup>¼</sup> 1, *<sup>q</sup>*, *<sup>q</sup>*2, *<sup>q</sup>*3,… ð Þ *<sup>q</sup>*<sup>&</sup>gt; <sup>1</sup> , ½ � *<sup>a</sup>*, *<sup>b</sup>* <sup>∪</sup> f g *<sup>c</sup>*, *<sup>d</sup>* where *a*<*b* and *a*, *b*,*c*, *d*∈ , and the Cantor set. It therefore contains the popular cases of the continuous, the discrete, and the quantum calculus.

Operators that aid the description of a time scale are the "forward jump operator", denoted by *σ*ð Þ*t* , the "backward jump operator", denoted by *ρ*ð Þ*t* , and the "graininess function", denoted by *μ*ð Þ*t* . These operators are defined for *t*∈ as

$$\sigma(t) \coloneqq \inf \left\{ s \in \mathbb{T} : s > t \right\}, \quad \rho(t) \coloneqq \sup \{ s \in \mathbb{T} : s < t \}, \quad \mu(t) \coloneqq \sigma(t) - t. \tag{1}$$

Since is closed, *σ*, *ρ* : ! and *μ* : ! ½0*;*∞Þ. **Table 1** provides values of the corresponding operators for different examples of time scales.


#### **Table 1.**

*The description of the time scales functions σ*, *ρ*, *μ for the examples of , , and q*<sup>0</sup> *(q*>1*).*

Using these operators, any *t*∈ can be classified as:


*Dynamic Equations on Time Scales DOI: http://dx.doi.org/10.5772/intechopen.104691*

We say that a point *t*∈ is isolated, if it is right- and left-scattered. We say that a point *t*∈ is dense, if it is right- and left-dense. Note that for ¼ , every point is dense and, for ¼ , every point is isolated.

**Example 2.1.** El Nino events can be described using a time scale. El Nino events between 2002 and 2017 have been observed in the time intervals 2002–2003, 2004–2005, 2006–2007, 2009–2010, and 2014–2016 [4], which suggests the corresponding time scale (**Figure 1**, **Table 2**)

$$\mathbb{T} = \cup\_{i=0}^{5} [a\_i, a\_{i+1}]$$

with ð*a*0, *a*1, *a*2, … , *a*5Þ ¼ ð Þ 2002, 2004, 2006, 2009, 2014, 2015 .


**Figure 1.**

*Part of the time line containing points in the time scale . Curly lines identify intervals within . Here, t*<sup>1</sup> ∈ ð Þ 2004, 2005 *, t*<sup>2</sup> *is the last point in the interval* ½ � 2004, 2005 *, and t*<sup>3</sup> ¼ 2006 *is the first point in* ½ � 2006, 2007 *.*


**Table 2.**

*The functions σ*, *ρ*, *μ for the time points t*1, *t*2, *t*<sup>3</sup> ∈ *based on Figure 1.*

The following notation is commonly used for *t*∈,

$$\boldsymbol{\sigma}^{\boldsymbol{n}}(t) = \underbrace{(\boldsymbol{\sigma}\circ\boldsymbol{\sigma}\circ\ldots\circ\boldsymbol{\sigma})}\_{n-\text{times}}(t), \qquad \qquad \boldsymbol{\rho}^{\boldsymbol{n}}(t) = \underbrace{(\boldsymbol{\rho}\circ\boldsymbol{\rho}\circ\ldots\circ\boldsymbol{\rho})}\_{n-\text{times}}(t).$$

#### **2.1 Functions on time scales**

We can now consider scalar functions on time scales, that is, *f* : ! , and discuss their properties. We define the subset *<sup>κ</sup>* as follows: If has a left-scattered maximum *<sup>m</sup>* <sup>∈</sup>, then *<sup>κ</sup>* <sup>¼</sup> nf g *<sup>m</sup>* , else *<sup>κ</sup>* <sup>¼</sup> .

**Definition 2.** *<sup>f</sup>*: ! is called regressive, if, for all *<sup>t</sup>*∈*<sup>κ</sup>* ,

$$1 + \mu(t)f(t) \neq 0$$

and is called positively regressive, if, for all *t*∈*<sup>κ</sup>* ,

$$1 + \mu(t)f(t) > 0.$$

The following are properties of *f* : ! that later identify integrability.

**Definition 3.** *f* : ! is called regulated provided its right-sided limit exists (as a finite value) for all right-dense points and its left-sided limit exists (as a finite value) for all left-dense points.

Even though every regulated function on a compact interval is bounded, in general, max *<sup>a</sup>*≤*t*≤*<sup>b</sup> f t*ð Þ and min *<sup>a</sup>* <sup>≤</sup>*t*≤*<sup>b</sup> f t*ð Þ do not need to exist for regulated *f* : ! .

**Definition 4.** *f* : ! is called rd-continuous if *f* is continuous at all right-dense points and its left-sided limit exists (as a finite value) for all left-dense points. The set of rd-continuous functions is denoted by *Crd* ¼ *Crd*ð Þ¼ *Crd*ð Þ , .

Note that, if *f* : ! is continuous, then *f* is rd-continuous. If *f* is rd-continuous, then *f* is regulated.

The set of rd-continuous and regressive (positively regressive) functions is denoted by R ¼ Rð Þ¼ Rð Þ , (R<sup>þ</sup> ¼ Rþð Þ¼ Rþð Þ , ).

Beside the classical addition and subtraction of functions, time scales calculus introduces the so-called "circle plus", denoted by ⊕ , and "circle minus", denoted by ⊖ . These operations are defined for *f*, *g* : ! as follows

$$\begin{aligned} (f \oplus \mathbf{g})(t) &= f(t) + \mathbf{g}(t) + (\mu \mathbf{f} \mathbf{g})(t), \\ \text{and,} \quad \text{for } \mathbf{g} \in \mathcal{R}, \qquad (f \ominus \mathbf{g})(t) &= \frac{f(t) - \mathbf{g}(t)}{\mathbf{1} + (\mu \mathbf{g})(t)}. \end{aligned}$$

A useful property is that if *f*, *g* ∈ R (Rþ), then *f* ⊕ *g*, *f*⊖ *g* ∈ R (Rþ) implying that the (positively) regressive property is being carried over. Furthermore, ð Þ R, ⊕ forms an Abelian group with the inverse elements of *f* ∈ R given by ⊖ *f*.

For ¼ , the operators ⊕ and ⊖ correspond to the classical addition and subtraction.

#### **2.2 Differentiation**

**Definition 5.** Let *<sup>f</sup>* : ! and *<sup>t</sup>*∈*<sup>κ</sup>* . If there exists *f* <sup>Δ</sup>ð Þ*<sup>t</sup>* <sup>∈</sup> such that for all *ε*>0, there exists *δ*>0 such that

$$|f(\sigma(t)) - f(s) - f^\Delta(t)(\sigma(t) - s)| \le \varepsilon |\sigma(t) - s| \quad \text{for all} \quad s \in (t - \delta, t + \delta) \cap \mathbb{T},$$

then we call *f* <sup>Δ</sup>ð Þ*<sup>t</sup>* the delta (or Hilger) derivative of *<sup>f</sup>* at *<sup>t</sup>* <sup>∈</sup>*<sup>κ</sup>*

If *f* <sup>Δ</sup>ð Þ*<sup>t</sup>* exists for all *<sup>t</sup>*<sup>∈</sup> *<sup>κ</sup>* , we say that *f* is delta differentiable (or short: differentiable) and the function *f* <sup>Δ</sup> : ! is called delta derivative of *<sup>f</sup>* on *<sup>κ</sup>* .

.

If *f* is differentiable at *t*∈*<sup>κ</sup>* , then

$$f(\sigma(t)) = f(t) + \mu(t)f^{\Delta}(t).$$

The following notations are used equivalently

$$((f^\sigma)(t) = (f \circ \sigma)(t) = f(\sigma(t))\dots$$

The definition of a delta derivative can be extended to consider higher order derivatives. We say that *f* is twice delta differentiable with the second (delta) derivative *f* ΔΔ, if *f* <sup>Δ</sup> is (delta) differentiable on *<sup>κ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>κ</sup>* ð Þ*<sup>κ</sup>* .

*Dynamic Equations on Time Scales DOI: http://dx.doi.org/10.5772/intechopen.104691*

Note that the definition of delta derivatives focuses on the change forward in time. A corresponding definition that focuses on the change backward in time is referred to as nabla derivative, see for example [5].

**Theorem 2.2.** [See [3, Theorem 1.16]] Let *<sup>f</sup>* : ! and *<sup>t</sup>*∈*<sup>κ</sup>* . Then, the following holds:

i. If *t* is right-dense, then

$$f^{\Delta}(t) = \lim\_{s \to t} \frac{f(t) - f(s)}{t - s},$$

provided that the limit exists (as a finite number).

ii. If *f* is continuous at the right-scattered point *t*, then

$$f^{\Delta}(t) = \frac{f(\sigma(t)) - f(t)}{\mu(t)}.$$

Applying Theorem 2.2 for the case of ¼ , shows that the delta derivative is consistent with the classical derivative, that is, *f* <sup>Δ</sup>ðÞ¼ *<sup>t</sup> f t*ð Þ for *<sup>t</sup>*<sup>∈</sup> <sup>¼</sup> . For <sup>¼</sup> , the delta derivative collapses to the forward Euler operator, widely accepted as the discrete analogue of a derivative, that is, *f* <sup>Δ</sup>ðÞ¼ *<sup>t</sup> f t*ð Þ� <sup>þ</sup> <sup>1</sup> *f t*ð Þ if <sup>¼</sup> (see **Table 3**).


**Table 3.**

*Derivatives for the examples of* <sup>¼</sup> *,* <sup>¼</sup> *, and* <sup>¼</sup> *qN*<sup>0</sup> *(q*>1*). Note that* <sup>Δ</sup>*f t*ðÞ¼ *f t*ð Þ� <sup>þ</sup> <sup>1</sup> *f t*ð Þ *is the forward Euler operator.*

As in the continuous case, the differential operator is linear, that is, for *α*, *β* ∈ , *t*∈*<sup>κ</sup>* , and for (delta) differentiable functions *f*, *g* : ! ,

$$(af + \beta \mathbf{g})^{\Delta}(t) = af^{\Delta}(t) + \beta \mathbf{g}^{\Delta}(t).$$

The analogues of the product and the quotient rule on time scales take on slightly different forms. For (delta) differentiable functions *<sup>f</sup>*, *<sup>g</sup>* : ! , and *<sup>t</sup>*∈*<sup>κ</sup>* ,

$$(f\mathbf{g})^{\Delta}(t) = f^{\Delta}(t)\mathbf{g}^{\sigma}(t) + f(t)\mathbf{g}^{\Delta}(t) = f^{\Delta}(t)\mathbf{g}(t) + f^{\sigma}(t)\mathbf{g}^{\Delta}(t)$$

and, for *g t*ð Þ, *<sup>g</sup><sup>σ</sup>*ð Þ*<sup>t</sup>* 6¼ 0,

$$\left(\frac{f}{\mathcal{g}}\right)^{\Delta}(t) = \frac{f^{\Delta}(t)\mathbf{g}(t) - f(t)\mathbf{g}^{\Delta}(t)}{\mathbf{g}(t)\mathbf{g}^{\sigma}(t)}.$$

For ¼ , we have *f <sup>σ</sup>* <sup>¼</sup> *<sup>f</sup>* and *<sup>g</sup><sup>σ</sup>* <sup>¼</sup> *<sup>g</sup>* so that the classical product and quotient rule are retrieved. In the case of ¼ , we have the correspondent rules consistent with [6], namely

$$
\Delta(f\mathbf{g})(t) = (\Delta f(t))\mathbf{g}(t+\mathbf{1}) + f(t)(\Delta \mathbf{g}(t)) \\
= (\Delta f(t))\mathbf{g}(t) + f(t+\mathbf{1})(\Delta \mathbf{g}(t)).
$$

If *g t*ð Þ, *g t*ð Þ þ 1 6¼ 0, then

$$
\Delta \left( \frac{f(t)}{\mathbf{g}(t)} \right) = \left( \frac{f}{\mathbf{g}} \right)^{\Delta}(t) = \frac{(\Delta f(t))\mathbf{g}(t) - (\Delta \mathbf{g}(t))f(t)}{\mathbf{g}(t)\mathbf{g}(t+1)}.
$$

The modifications in the product and quotient rule highlight that some of the well established differentiation rules only carry over to time scales calculus after some adjustments. In fact, the product rule on time scales reveals that the useful property of power functions *f t*ðÞ¼ *t <sup>n</sup>* for *n*∈ <sup>0</sup> is no longer the simple reduction of the power by one, because

$$\left(t^2\right)^\Delta = (t\cdot t)^\Delta = t + \sigma(t),$$

which may not be delta differentiable. This indicates already that the series representation of functions requires further thought.

Also, considering the chain rule, we note that for ¼ ,

$$
\Delta(f \circ f)(t) = f^{\sigma}(t)f^{\Delta}(t) + f(t)f^{\Delta}(t) \\
= f^{\Delta}(t)(f(t) + f^{\sigma}(t)) \neq \mathcal{Y}(t)f^{\Delta}(t),
$$

for *f σ* ð Þ*t* 6¼ *f t*ð Þ. Thus, the powerful chain rule, often utilized in solving differential equations via a variable transformation, does not apply on time scales. In an attempt to generalize the chain rule for functions on time scales a few identities have been formulated. The next theorem provides such an expression based on works in [7, 8]. Other formulations can be found in [3].

**Theorem 2.3**. (See [3, Theorem 1.90]). Let *f* : ! be continuously differentiable and suppose *g* : ! is (delta) differentiable. Then *f*∘*g* : ! is (delta) differentiable and

$$(\int \!\!\!g)^{\Delta}(t) = \left\{ \int\_0^1 f'(\!\!\!g(t) + h\mu(t)\!\!g^{\Delta}(t)) \,\text{d}h \right\} \!\!g^{\Delta}(t).$$

An interesting observation is that the operators, Δ and *σ*, do generally not commute, that is, *f* <sup>Δ</sup> � �*<sup>σ</sup>* 6¼ *<sup>f</sup> <sup>σ</sup>* ð Þ<sup>Δ</sup>. Take for example <sup>¼</sup> *<sup>q</sup>*<sup>0</sup> with *<sup>q</sup>*>1, then

$$\left(\left(f^{\Delta}\right)^{\sigma}(t) = \frac{f(q^{2}t) - f(qt)}{\mu(qt)} \neq \frac{f(q^{2}t) - f(qt)}{\mu(t)} = (f^{\sigma})^{\Delta}(t),$$

since *μ*ð Þ¼ *qt qt q*ð Þ � 1 6¼ *t q*ð Þ¼ � 1 *μ*ð Þ*t* .

#### **2.3 Integration**

**Definition 6.** A continuous function *f* : ! is called pre-differentiable with (region of differentiation) *D*, provided that *D* ⊂ *<sup>κ</sup>* , *<sup>κ</sup>* n*D* is countable and contains no right-scattered elements of , and *f* is (delta) differentiable at each *t*∈ *D*.

**Theorem 2.4**. (See [3, Theorem 1.70]). Let *f* : ! be regulated. Then there exists a function *F* : ! which is pre-differentiable with region of differentiation *D* such that

$$F^\Delta(t) = f(t) \qquad \text{for all } t \in D.$$

The function *F* is called an pre-antiderivative of *f t*ð Þ.

If *<sup>F</sup>*<sup>Δ</sup>ðÞ¼ *<sup>t</sup> f t*ð Þ for all *<sup>t</sup>*∈*<sup>κ</sup>* , then *F* is called antiderivative of *f*.

We define the indefinite integral of a regulated function *f* by Ð *f t*ð ÞΔ*t* ¼ *F t*ðÞþ *C*, where *C* ∈ is an arbitrary integration constant and *F* is a pre-antiderivative of *f*. The Cauchy integral is defined by Ð *<sup>b</sup> <sup>a</sup> f t*ð ÞΔ*t* ¼ *F b*ð Þ� *F a*ð Þ for all *a*, *b*∈.

**Theorem 2.5**. (See [3, Theorem 1.74]). Every rd-continuous function *f* has an antiderivative. In particular, if *t*<sup>0</sup> ∈, then *F* defined by

$$F(t) \coloneqq \int\_{t\_0}^t f(s) \, \Delta s \qquad \text{for all } t \in \mathbb{T}$$

is an antiderivative of *f*.

For ¼ , the integral is consistent with the Rieman integral (see **Table 4**).


#### **Table 4.**

*Integrals for the examples of* <sup>¼</sup> *,* <sup>¼</sup> *, and* <sup>¼</sup> *qN*<sup>0</sup> *(q*>1*), and isolated time scales* I*, for which all points in* <sup>I</sup> *are assumed to be isolated. In all cases, s*, *<sup>t</sup>* <sup>∈</sup> *and s* <sup>&</sup>lt;*t. In the case of* <sup>¼</sup> *<sup>q</sup>*<sup>0</sup> *, we assume t* <sup>¼</sup> *qks.*

The integral operator is linear so that for *f*, *g* ∈*Crd* and *a*< *b*, *a*, *b*∈, and *α*, *β* ∈ ,

$$\int\_{a}^{b} (af + \beta \mathbf{g})(s) \, \Delta s = a \int\_{a}^{b} f(s) \, \Delta s + \beta \int\_{a}^{b} \mathbf{g}(s) \, \Delta s.$$

With the definition of integration on time scales, we have the machinery to introduce a series representation for time scales functions. In [9], see also [3], a time scales analogue of polynomials that allows a corresponding Taylor series expression was introduced using the recursive formulation

$$\lg\_0(t,s) = h\_0(t,s) \equiv \mathbf{1} \qquad \text{for all } t, s \in \mathbb{T}, s$$

and, for every *k*∈ 0,

$$\mathcal{g}\_{k+1}(t,s) = \int\_s^t \mathcal{g}\_k(\sigma(\tau), s) \, \Delta \tau \qquad \text{for all } s, t \in \mathbb{T}, s$$

and

$$h\_{k+1}(t,s) = \int\_s^t h\_k(\tau,s)\,\Delta\tau \qquad \text{for all } s, t \in \mathbb{T}.$$

Now, *h*<sup>Δ</sup> *<sup>k</sup>* ð Þ¼ *t*, *s hk*ð Þ *t*, *s* and *g*<sup>Δ</sup> *<sup>k</sup>* ð Þ¼ *<sup>t</sup>*, *<sup>s</sup> gk*ð Þ <sup>σ</sup>ð Þ*<sup>t</sup> ;<sup>s</sup>* for *<sup>k</sup>*<sup>∈</sup> and *t,s <sup>T</sup>*<sup>κ</sup> . Two Taylor series representations can be formulated for a time scales function f, one that uses the time scales polynomials *gk* and one that uses the polynomials *hk*, see Section 1.6 in [3] for more details.

#### **3. Linear dynamic equations**

This chapter provides a brief introduction to first order dynamic equations and provides a selected summary of [3], extended by applications. A first order dynamic equation is of the form

$$y^{\Delta}(t) = f(t, y, y^{\sigma}),\tag{2}$$

for *<sup>y</sup>* : ! *<sup>n</sup>* and *<sup>f</sup>* : � *<sup>n</sup>* � *<sup>n</sup>* ! *<sup>n</sup>* with *<sup>n</sup>*<sup>∈</sup> <sup>1</sup> <sup>¼</sup> f g 1, 2, 3, … . A first order initial value problem (short: IVP) is then given by (2) with an initial condition *y t*ð Þ¼ <sup>0</sup> *<sup>y</sup>*<sup>0</sup> <sup>∈</sup> *<sup>n</sup>* for *<sup>t</sup>*<sup>0</sup> <sup>∈</sup>. A function *<sup>y</sup>* : ! *<sup>n</sup>* is called a *solution* of (2) if *<sup>y</sup>* satisfies the equation for all *t*∈*<sup>κ</sup>* .

We call (2) linear if

$$f(t, y, y^{\sigma}) = f\_1(t)y + f\_2(t), \text{ or } f(t, y, y^{\sigma}) = f\_1(t)y^{\sigma} + f\_2(t), \text{ or }$$

where *<sup>f</sup>* <sup>1</sup>, *<sup>f</sup>* <sup>2</sup> : ! *<sup>n</sup>*. We say the linear dynamic equation is homogeneous, if *<sup>f</sup>* <sup>2</sup> � 0.

#### **3.1 Scalar case**

We first focus on the scalar case of (2), that is, *f* : ! . Based on the above definition of linearity, there are two forms a linear, homogeneous, first order dynamic equation can have:

$$
\mathbf{y}^{\Delta} = p(t)\mathbf{y},\tag{3}
$$

$$y^{\Delta} = p(t)y^{\sigma}, \text{ for } p \colon \mathbb{T} \to \mathbb{R} \tag{4}$$

Note that for <sup>¼</sup> , *<sup>y</sup><sup>σ</sup>* <sup>¼</sup> *<sup>y</sup>* and therefore *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> *p t*ð Þ*y<sup>σ</sup>* <sup>¼</sup> *p t*ð Þ*<sup>y</sup>* so that both, (3) and (4), are the time scales analogues of *y*<sup>0</sup> ¼ *p t*ð Þ*y*.

If *p* ∈ R, then (3) is called *regressive* and if �*p* ∈ R, then (4) is called *regressive*.

The unique solution to (3) with initial condition *y t*ð Þ¼ <sup>0</sup> 1 for some *t*<sup>0</sup> ∈ is denoted by *y t*ðÞ¼ *ep*ð Þ *t*, *t*<sup>0</sup> and is called the time scales exponential function. The unique solution to (4) with initial condition *y t*ð Þ¼ <sup>0</sup> 1 is *y t*ðÞ¼ *e*<sup>⊖</sup> ð Þ �*<sup>p</sup>* ð Þ *t*, *t*<sup>0</sup> .

**Table 5** contains the time scales analogues of the exponential function for the dense time scale ¼ , the discrete time scale ¼ , and the quantum time scale <sup>¼</sup> *<sup>q</sup>*<sup>0</sup> .


#### **Table 5.**

*The exponential function for the continuous, discrete, and quantum time scale (q*>1*), assuming p* ∈ R*.*

*Dynamic Equations on Time Scales DOI: http://dx.doi.org/10.5772/intechopen.104691*

The **Table 5** reveals a crucial aspect of the time scales exponential function, namely that the positivity property, known for the traditional exponential function, does not uphold on time scales. Take for example, ¼ and *p* ¼ �3, then *p* ∈ R as <sup>1</sup> <sup>þ</sup> *<sup>p</sup>* ¼ �<sup>2</sup> 6¼ 0, but *ep*ð Þ¼ � *<sup>t</sup>*, 0 ð Þ<sup>2</sup> *<sup>t</sup>* which is negative for odd values of *<sup>t</sup>*. If however *p*∈ Rþ, then *ep*ð Þ *t*, *t*<sup>0</sup> >0, restoring the positivity property. Note that if ¼ , then any function *p*∈ R<sup>þ</sup> since 1 þ *μ*ð Þ*t p t*ðÞ¼ 1>0.

Some of the properties of the time scales exponential function are consistent with the convenient properties in the continuous case. If *p*, *q*∈ R and *t*, *s* ∈, then

$$\begin{aligned} \text{i. } &e\_0(t,s) = \mathbf{1}, e\_p(t,t) = \mathbf{1}, \\\\ \text{ii. } &e\_{p \oplus\_q q}(t,s) = e\_p(t,s)e\_q(t,s), \\\\ \text{iii. } &e\_{\ominus\_p p}(t,s) = e\_p(s,t) = \frac{1}{e\_p(t,s)}, \\\\ \text{iv. } &e\_p(t,r)e\_p(r,s) = e\_p(t,s), \\\\ \text{v. } &e\_p(\sigma(t),s) = (\mathbf{1} + \mu(t)p(t))e\_p(t,s). \end{aligned}$$

**Theorem 3.1.** [See [3, Theorem 2.39]] If *p* ∈ R and *a*, *b*,*c*∈, then

$$\begin{aligned} \int\_a^b p(t)e\_p(t,c)\,\Delta t &= e\_p(b,c) - e\_p(a,c) \\ \int\_a^b p(t)e\_p(c,\sigma(t))\,\Delta t &= e\_p(c,a) - e\_p(c,b) .\end{aligned}$$

As an application of linear, homogeneous, first order dynamic equations, one may consider the Malthusian growth model. In "An essay on the principle of population" from 1798, Thomas Robert Malthus proposed an exponential law of population growth with the corresponding differential equation

$$P' = rP, \quad P(t\_0) = P\_{0,\*}$$

where *P* is the population at time *t*, *r* is the inherent growth rate, and *P*<sup>0</sup> is the initial population level at time *t*<sup>0</sup> ∈ . This linear, homogeneous, first order differential equation has the solution *P t*ðÞ¼ *<sup>e</sup>r t*ð Þ �*t*<sup>0</sup> *<sup>P</sup>*0. Assuming a positive initial population level *<sup>P</sup>*<sup>0</sup> <sup>&</sup>gt;0, it follows that for a positive growth rate *r*>0, the population increases exponentially. If instead *<sup>r</sup>*<0 and *<sup>P</sup>*<sup>0</sup> <sup>&</sup>gt;0, then the population goes extinct as lim *<sup>t</sup>*!<sup>∞</sup>*er t*ð Þ �*t*<sup>0</sup> *<sup>P</sup>*<sup>0</sup> <sup>¼</sup> 0. Despite its simplicity and the unrealistic behavior of unbounded population levels for r > 0, the Malthusian model can sometimes serve short-term predictions.

Let us now consider the corresponding time scales model (3) with initial condition *P t*ð Þ¼ <sup>0</sup> *<sup>P</sup>*<sup>0</sup> <sup>&</sup>gt; 0 and inherent growth rate *<sup>r</sup>*>0, that is, *<sup>P</sup>*<sup>Δ</sup> <sup>=</sup> *rP* with *P t*ð Þ¼ <sup>0</sup> *<sup>P</sup>*<sup>0</sup> for *<sup>t</sup>*<sup>0</sup> <sup>∈</sup>. The respective solution is then *P t*ðÞ¼ *er*ð Þ *t*, *t*<sup>0</sup> *P*0, which is unbounded for *r;P*<sup>0</sup> >0, see **Figure 2**. Thus, for *r;P*<sup>0</sup> >0, the behavior of the solution is consistent with the solution in the continuous case. However, for *r*<0, the population does not have to go extinct but can result in biologically unmeaningful behavior as solutions can become negative.

Using the time scales exponential function that solves a linear, homogeneous, first order dynamic equation, we can use the variation of constants formula to obtain the solution to a linear, nonhomogeneous, first order dynamic equation.

#### **Figure 2.**

*The behavior of the solution to P*<sup>Δ</sup> <sup>¼</sup> *rP with P t*ð Þ¼ <sup>0</sup> *<sup>P</sup>*<sup>0</sup> *where r = 0.45, t0 = 1, and P0 = 0.1, for* <sup>¼</sup> *,* <sup>¼</sup> *and* <sup>¼</sup> <sup>1</sup>*:*3<sup>ℕ</sup><sup>0</sup> *. The solid line represents the solution in the continuous case, the open circle represents the solution in the discrete case, and the stars represent the solution in the quantum calculus case with q=1.3.*

**Theorem 3.2.** [See [3, Theorems 2.74 & 2.77]] Suppose *p*∈ R, *f* ∈*Crd; t*<sup>0</sup> ∈ and *y*<sup>0</sup> ∈ then the unique solution to

$$\mathcal{Y}^{\Delta} = p(t)\mathcal{Y} + f(t), \qquad \qquad \mathcal{Y}(t\_0) = \mathcal{Y}\_0$$

is given by

$$y(t) = e\_p(t, t\_0)y\_0 + \int\_{t\_0}^t e\_p(t, \sigma(s))f(s)\,\Delta s.$$

Furthermore, the unique solution to

$$y^\Delta = -p(t)y^\sigma + f(t), \qquad \mathcal{y}(t\_0) = \mathcal{y}\_0$$

is given by

$$y(t) = e\_{\ominus p}(t, t\_0) y\_0 + \int\_{t\_0}^t e\_{\ominus p}(t, s) f(s) \, \Delta s.t$$

**Example 3.3.** Suppose that the life span of a certain species is one time unit. Suppose that just before the species dies out, eggs are laid that are hatch after one time unit. The species is therefore only alive on <sup>¼</sup> <sup>∪</sup> <sup>∞</sup> *<sup>k</sup>*¼<sup>0</sup>½ � <sup>2</sup>*k*, 2*<sup>k</sup>* <sup>þ</sup> <sup>1</sup> , see also [3, Example 1.39] and [10]. Suppose further that during the specie's, life cycle, the species reduces due to external factors with rate *d* ∈ (0, 1) and at the end of the life cycle *t* ¼ 2*k* þ 1, the individuals alive in (2*k*, 2*k* þ 1) lay eggs that result in the reproduction rate *r*>0. The corresponding dynamic equation for the species *N*(*t*) at time *t*, is then

$$N^{\Delta}(t) = p(t)N(t), \qquad \text{with} \quad p(t) = \begin{cases} -d & t \in [2k, 2k+1) \\ r & t = 2k+1 \end{cases}$$

and initial condition *N*ð Þ¼ 0 *N*0. We note that even though *p t*ð Þ is discontinuous at *t* ¼ 2*k* þ 1, *p t*ð Þ∈ R. Theorem 3.2 gives the population at time *t*∈ ½ � 2*m*, 2*m* þ 1 as

*Dynamic Equations on Time Scales DOI: http://dx.doi.org/10.5772/intechopen.104691*

$$\begin{split} N(t) &= N\_0 e\_p(t, t\_0) = N\_0 e\_p(t, 2m) \prod\_{k=0}^{m-1} \left( e\_p(2k+1, 2k) e\_p(2k+2, 2k+1) \right) \\ &= N\_0 \exp\left\{ \int\_{2m}^t -d \, \text{ds} \right\} \left\{ \prod\_{k=0}^{m-1} \exp\left\{ \int\_{2k}^{2k+1} -d \, \text{ds} \right\} (1+r) \right\} = N\_0 e^{-d(t-m)} (1+r)^m .\end{split}$$

**Example 3.4.** Newton's law of cooling suggests that the temperature of an object at time *t*, *T t*ð Þ, changes dependent on the temperature of its surrounding, *Tm*. Then, *T*<sup>0</sup> ðÞ¼� *t κ*ð Þ *T* � *Tm* , where *κ* is the heat transfer coefficient. Suppose that an object with initial temperature *T*<sup>0</sup> is cooled in a lab environment. Due to safety regulations, once the lab assistant leaves the work space, the object can only be exposed to an environment that preserves the current temperature of the object. The cooling of the object can be modeled using time scales with the underlying time domain to be the working hours of the lab assistant. Assume that the lab assistant's working hours, and therefore the time scale, is of the form <sup>¼</sup> <sup>∪</sup> <sup>∞</sup> *<sup>i</sup>*¼<sup>0</sup> *ai*, *bi* ½ � ∪ *ci*, *di* ½ �, where the interval *ai*, *bi* ½ � are the working hours prior to lunch, and *ci*, *di* ½ � are the working hours of the lab assistant after lunch of day *i*. One way of modeling this scenario on time scales is

$$T^{\Delta} = -p(t)(T - T\_m), \qquad p(t) = \begin{cases} \kappa & t \in [a\_i, b\_i) \cup [c\_i, d\_i) \\ \mathbf{0} & t \in \{b\_i, d\_i\} \end{cases}$$

with initial temperature *T t*ð Þ¼ <sup>0</sup> *T*<sup>0</sup> for *t*<sup>0</sup> ∈. Since *p t*ð Þ is rd-continuous and regressive, the theorems above can be applied despite the discontinuity of *p t*ð Þ.

**Example 3.5.** The following example is from [11], where a Keynesian-Cross model with lagged income is considered. Here, the aggregated income *y* changes according to

$$\mathcal{Y}^{\Delta} = \delta[d^{\sigma}(t) - \mathcal{y}], \qquad t \ge t\_0 \in \mathbb{T},$$

where *d t*ð Þ is the aggregated demand at time *t* and *δ*∈ð Þ 0, 1 is the "adjustment speed". Since *d t*ð Þ can be expressed as the addition of aggregated consumption (c), aggregated investment (I), and governmental spending (G), we have *d t*ðÞ¼ *c t*ðÞþ *I* þ *G* for *I*, *G* ∈ð Þ 0, ∞ . Under the assumption that aggregated consumption is itself linear in the aggregated income, we have *c t*ðÞ¼ *a* þ *by t*ð Þ with *a*, *b*>0 so that the model reads as

$$\mathbf{y}^{\Delta} = \delta[\mathfrak{a} + b\mathfrak{y}^{\sigma} + I + \mathbf{G} - \mathfrak{y}].$$

Under the assumption that *p t*ð Þ: <sup>¼</sup> <sup>1</sup> � *<sup>δ</sup>bμ*ð Þ*<sup>t</sup>* 6¼ 0, we can apply *<sup>y</sup><sup>σ</sup>* <sup>¼</sup> *<sup>y</sup>* <sup>þ</sup> *<sup>μ</sup>y*<sup>Δ</sup>, and express the dynamic equation as

$$\mathcal{y}^{\Delta} = \frac{\delta(a+I+G)}{p(t)} + \frac{\delta(b-1)}{p(t)}\mathcal{y}.$$

which is a linear, non-homogeneous, first order dynamic equation. It is left as an exercise to apply the techniques of this subsection to derive an explicit solution to this dynamic equation.

**Example 3.6.** Let us consider a time scales analogue of the popular logistic growth model *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> *ry* <sup>1</sup> � *<sup>y</sup> K* , namely,

$$
\gamma^{\Delta} = r\gamma^{\sigma} \left( 1 - \frac{\mathcal{Y}}{K} \right), \qquad \mathcal{y}(t\_0) = \mathcal{y}\_0,\tag{5}
$$

with growth rate *r* > 0, and carrying capacity *K* > 0, and initial population size *y t*ð Þ<sup>0</sup> > 0 at time *t*<sup>0</sup> ∈ . Even though this is an example of a nonlinear dynamic equation of first order, we can apply the substitution *<sup>z</sup>* <sup>¼</sup> <sup>1</sup> *y* for *y* 6¼ 0, to obtain the linear dynamic equation

$$z^{\Delta} = \frac{-\mathcal{Y}^{\Delta}}{\mathcal{Y}^{\sigma}} = -r\mathbf{z} + \frac{r}{K}, \qquad z(t\_0) = \frac{1}{\mathcal{Y}\_0}.$$

For �*r*∈ R, the solution is then given by Theorem 3.2. Using also Theorem 3.1 and resubstituting yields

$$\chi(t) = \frac{\wp\_0 K}{e\_{-r}(t, t\_0)(K - \wp\_0) + \wp\_0}. \tag{6}$$

,

It can be easily checked that *y t*ð Þ¼ <sup>0</sup> *y*<sup>0</sup> and that *y* solves (5), see also [12].

Note that for <sup>¼</sup> , (5) collapses to the Verhulst model *<sup>y</sup>*<sup>0</sup> <sup>¼</sup> *ry* <sup>1</sup> � *<sup>y</sup> K* and the solution (6) reads in this case as

$$\mathcal{Y}(t) = \frac{\mathcal{Y}\_0 K}{e^{-r(t-t\_0)} \left(K - \mathcal{Y}\_0\right) + \mathcal{Y}\_0}$$

which coincides with the classical solution.

#### **3.2 Linear systems**

Let us now consider (2) with *<sup>f</sup>*: � *<sup>n</sup>* � *<sup>n</sup>* ! *<sup>n</sup>* for *<sup>n</sup>* <sup>∈</sup> <sup>¼</sup> f g 1, 2, 3, … . In order to extend the solution methods for linear first order dynamic equations that were introduced in the previous section for scalar functions, the definitions of rd-continuity and delta differentiability have to be first extended to matrix valued functions *<sup>A</sup>* : ! *<sup>m</sup>*�*<sup>n</sup>*. This adjustment is mostly proposed element-wise. More precisely, *A* is rd-continuous on if *aij* is rd-continuous on for all 1≤*i*≤ *n*, 1≤*j*≤ *m*. The class of all such rd-continuous *<sup>m</sup>* � *<sup>n</sup>*-matrix-valued functions on is then denoted by *Crd* , *<sup>m</sup>*�*<sup>n</sup>* ð Þ. Similarly, we say that *A* is delta differentiable (or short: differentiable), if *aij* is delta differentiable for all 1≤*i* ≤*n*, 1≤ *j*≤ *m*. Similar to the scalar case, the following identity holds for any matrixvalued (delta) differentiable function *A*,

$$A^{\sigma}(t) = A(t) + \mu(t)A^{\Delta}(t).$$

The property of regressive is however not defined elementwise. Instead, we say that *<sup>A</sup>* <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is regressive if *In* <sup>þ</sup> *<sup>μ</sup>*ð Þ*<sup>t</sup> A t*ð Þ is invertible for all *<sup>t</sup>*<sup>∈</sup> *<sup>κ</sup>* , where *In* ∈ *<sup>n</sup>*�*<sup>n</sup>* is the identity matrix. The class of rd-continuous and regressive functions is denoted by <sup>R</sup> , *<sup>n</sup>*�*<sup>n</sup>* ð Þ (or short <sup>R</sup>).

Note that even if all entries of *A* are regressive, *A* does not have to be regressive. Take for example ¼ with

$$A = \begin{bmatrix} a\_{11} & a\_{12} \\ a\_{21} & a\_{22} \end{bmatrix} = \begin{bmatrix} \mathbf{0} & -2 \\ -2 & \mathbf{3} \end{bmatrix}.$$

Then all entries are regressive as 1 þ *aij* 6¼ 0 for all 1≤*i*, *j*≤2 but detð Þ¼ *I* þ *A* 0. As for the scalar case, differentiation is linear, that is,

$$(aA + \beta B)^{\Delta}(t) = aA^{\Delta}(t) + \beta B^{\Delta}(t)$$

for differentiable *m* � *n*-matrix-valued functions *A*, *B*, and *α*, *β* ∈ . We consider

$$\mathbf{y}^{\Delta} = \mathbf{A}(t)\mathbf{y} \tag{7}$$

to be the system analogue of (3). If *A* is *n* � *n* matrix valued function, then, the unique solution to (7) with *y*ð*t*0Þ ¼ *In*, where *In* is the *n* � *n* identity matrix, is denoted by *y t*ðÞ¼ *eA*ð Þ *<sup>t</sup>*, *<sup>t</sup>*<sup>0</sup> . If *<sup>A</sup>* <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and <sup>¼</sup> then *eA*ð Þ¼ *<sup>t</sup>*, *<sup>t</sup>*<sup>0</sup> *<sup>e</sup>A t*ð Þ –*t*<sup>0</sup> , and if <sup>¼</sup> , then *eA*ð Þ¼ *<sup>t</sup>*, *<sup>t</sup>*<sup>0</sup> ð Þ *<sup>I</sup>* <sup>þ</sup> *<sup>A</sup> <sup>t</sup>*–*t*<sup>0</sup> . The analogue of (4) in higher dimensions is

> *<sup>y</sup>*<sup>Δ</sup> ¼ �*A*<sup>∗</sup> ð Þ*<sup>t</sup> <sup>y</sup><sup>σ</sup>*,

where *A*<sup>∗</sup> (*t*) is the conjugate transpose of *A*(*t*).

**Theorem 3.7.** (See [3, Theorems 5.24 & 5.27]). Let *<sup>A</sup>* <sup>∈</sup> <sup>R</sup> , *<sup>n</sup>*�*<sup>n</sup>*, *<sup>n</sup>*�*<sup>n</sup>* ð Þ and suppose that *<sup>f</sup>*: ! *<sup>n</sup>* is rd-continuous. Let *<sup>t</sup>*<sup>0</sup> <sup>∈</sup> and *<sup>y</sup>*<sup>0</sup> <sup>∈</sup> *<sup>n</sup>*. Then, the initial value problem

$$\mathcal{Y}^{\Delta} = A(t)\mathcal{Y} + f(t), \qquad \mathcal{Y}(t\_0) = \mathcal{Y}\_0.$$

is given by

$$
\varphi(t) = e\_A(t, t\_0)\mathcal{y}\_0 + \int\_{t\_0}^t e\_A(t, \sigma(\tau)) f(\tau) \,\Delta\tau.
$$

The unique solution to

$$\boldsymbol{y}^{\Delta} = -\boldsymbol{A}^\*(t)\boldsymbol{y}^{\sigma} + \boldsymbol{f}(t), \qquad \boldsymbol{\mathcal{Y}}(t\_0) = \boldsymbol{\mathcal{Y}}\_0$$

is given by

$$y(t) = e\_{\ominus A^\circ}(t, t\_0) y\_0 + \int\_{t\_0}^t e\_{\ominus A^\circ}(t, \tau) f(\tau) \,\Delta \tau.$$

**Example 3.8.** In [13], the authors consider the Cucker-Smale type model on an isolated (i.e., every *t* ∈ is isolated) with sup ¼ ∞ and supf g *μ*ð Þ*t* : *t* ∈ < ∞,

$$\begin{aligned} \boldsymbol{x}\_i^{\Delta} &= \boldsymbol{v}\_i \\ \boldsymbol{v}\_i^{\Delta} &= \frac{1}{N} \sum\_{j=1}^{N} a\_{ij} (\boldsymbol{v}\_j - \boldsymbol{v}\_i), \end{aligned} \tag{8}$$

where *aij* ∈ <sup>þ</sup> <sup>0</sup> ¼ ½ Þ 0, ∞ and *i* ∈f g 1, 2, … , *N* represents the impact of agent's *j* opinion onto the agent's *i* opinion. The variable *xi* represents the state of agent *i*, and *vi* is the consensus parameter of agent *i*. The original Cucker-Smale model, see [14], is a discrete time system discussing the flock behavior of birds, where *vi* represents the velocity of bird *i* and *xi* is its position. The weights *aij* quantify the way the birds influence each other.

Note that since is isolated, we can equivalently write (8) as

$$x\_i(\sigma(t)) = x\_i(t) + \mu(t)v\_i(t), \qquad v\_i(\sigma(t)) = v\_i(t) + \frac{\mu(t)}{N} \sum\_{j=1}^{N} a\_{ij}(v\_j(t) - v\_i(t)),$$

or in form of a system in *y* ¼ ð Þ *x*1, *x*2, … , *xN*, *v*1, *v*2, … , *vN T*,

$$\mathbf{y}^{\Delta} = B\mathbf{y}, \qquad B = \frac{\mathbf{1}}{N} \begin{bmatrix} \mathbf{0}\_N & N\mathbf{I}\_N \\ \mathbf{0}\_N & A - D, \end{bmatrix}, \tag{9}$$

where ð Þ *<sup>A</sup> ij* <sup>¼</sup> *aij* for *<sup>i</sup>*, *<sup>j</sup>*<sup>∈</sup> f g 1, 2, … , *<sup>N</sup>* , *<sup>D</sup>* <sup>¼</sup> *diag d*ð Þ 1, *<sup>d</sup>*2, … , *dN* with *dk* <sup>¼</sup> <sup>P</sup>*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup>*akj*, 0*<sup>N</sup>* is a matrix of dimension *N* � *N* with all entries being zero, and *IN* is the identity matrix of dimension *N* � *N*.

If *B* ∈ R, then the solution to (9) with initial condition *y*(*t*0) = *y*<sup>0</sup> is *y*(*t*) = *e*<sup>B</sup> (*t,t*0) *y*0. In order for *B*∈ R, *NI*<sup>N</sup> + *μ*(*t*)(*A-D*) must be invertible because

$$
\tilde{B}(t) = I\_{2N} + \mu(t)B = \begin{bmatrix} I\_N & \mu(t)I\_n \\ \mathbf{0}\_N & \mathbf{C}(t) \end{bmatrix}, \qquad \mathbf{C}(t) = I\_N + \mu(t)\frac{\mathbf{1}}{N}(A - D),
$$

and

$$\det(\tilde{B}(t)) = \det(I\_{2N} + \mu(t)B) = \det(I\_N)\det(C(t)).$$

We conclude this section by examples of nonlinear dynamic equations that can be transformed into a system of linear dynamic equations of first order, so that Theorem 3.7 provides its solution.

**Example 3.9.** Let be again an isolated time scale, that is, every point in is isolated and *inf* f g *μ*ð Þ*t* : *t* ∈ > 0. Consider

$$\alpha^{\sigma^k} = \frac{K\mathbf{x}}{(1 - \mu(t)a)K + \mu(t)a\mathbf{x}},\tag{10}$$

with initial values *x* ! <sup>0</sup> <sup>¼</sup> ð Þ *<sup>x</sup>*0, *<sup>x</sup>*1, … , *xk*�<sup>1</sup> <sup>∈</sup> ð Þ 0, <sup>∞</sup> *<sup>k</sup>* , *K* > 0, and *–α* ∈ þ. Eq. (10) is a delayed Beverton-Holt model and can be used to model mature individuals of a population, assuming that it takes *k* reproductive cycles for an individual to become mature, where the length of a reproductive cycle starting at *t* is *μ*ð Þ*t* . An application may be populations where the lengths between breeding cycles is temperature dependent. Model (10) has been considered in [15] (and, for ¼ , in [16]), where the authors applied the transformation *y* ≔ *<sup>K</sup> <sup>x</sup>* for *x* 6¼ 0 to obtain

$$Y^{\Delta} = A(t)Y + \mathbf{b}(t) \quad \text{with} \quad A(t) = \frac{\mathbf{1}}{\mu(t)} \begin{bmatrix} \mathbf{0}\_{k-1} & I\_{k-1} \\ -\mu a & -\mathbf{s} \end{bmatrix}, \quad \mathbf{b}(t) = \begin{pmatrix} \mathbf{0}\_{k-1} \\ a \end{pmatrix}, \tag{11}$$

where **<sup>s</sup>** <sup>¼</sup> *<sup>k</sup>* 1 � �, *<sup>k</sup>* 2 � �, *<sup>k</sup>* 3 � �, … , *<sup>k</sup> k* � 1 � � � � and **<sup>0</sup>***<sup>k</sup>*�<sup>1</sup> <sup>∈</sup> *<sup>k</sup>*�1�<sup>1</sup> is vector of zeros. Applying Theorem 3.7, to (11) yields the solution.

**Example 3.10.** In [17], the authors proposed the following nonlinear system of dynamic equations to model the spread of a contagious disease,

$$\begin{aligned} \mathcal{S}^\Delta &= -\beta(t)\mathcal{S}^\sigma I - \nu(t)\mathcal{S} + \chi(t)I + \nu(t)\kappa, \\ I^\Delta &= \beta(t)\mathcal{S}^\sigma I - \chi(t)I - \nu(t)I. \end{aligned}$$

In line with well-established epidemic models, the population was compartmentalized into susceptible *S* and infected *I* individuals. The model assumes that the disease is spread by contact with an infected individual with a transmission rate of *β* >0. The recovery rate is assumed to be *γ* >0 and recovered individuals rejoin the

group of susceptible individuals. The death rate is *ν*ð Þ*t* across the population and *ν*ð Þ*t κ* newborns join the group of susceptibles.

By introducing a new variable *w* ≔ *S* þ *I*, *w*<sup>Δ</sup> ¼ �*ν*ð Þ*t w* þ *ν*ð Þ*t κ*. This first order, linear, nonhomogeneous dynamic equation can be solved using Theorem 3.2, assuming �*ν*ð Þ*t* ∈ R. The solution is then *w t*ðÞ¼ *e*�*<sup>ν</sup>*ð Þ *t*, *t*<sup>0</sup> ð Þþ *I*<sup>0</sup> þ *S*<sup>0</sup> � *κ κ*, so that, after recalling that *S* ¼ *w t*ð Þ� *I*, the dynamic equation in *I* can be expressed as

$$I^{\Delta} = \beta(t)(\omega^{\sigma} - I^{\sigma})I - \gamma(t)I - \nu(t)I.$$

Although the dimension has been reduced to one, the dynamic equation is still nonlinear. Defining however *<sup>y</sup>* <sup>¼</sup> <sup>1</sup> *<sup>I</sup>* for *I* 6¼ 0 yields again a linear dynamic equation, namely

$$\mathcal{Y}^\Delta = (-\beta(t)\nu^\sigma(t) + \gamma(t) + \nu(t))\mathcal{Y}^\sigma + \beta(t).$$

Applying Theorem 3.2 gives the solution

$$y(t) = e\_{\ominus\_{\mathcal{P}}p}(t, t\_0)y\_0 + \int\_{t\_0}^{t} e\_{\ominus\_{\mathcal{P}}p}(t, s)\beta(s)\,\Delta s, t$$

where *p t*ðÞ¼ *β*ð Þ*t w*ð Þ� *σ*ð Þ*t* ð Þ *γ*ðÞþ*t ν*ð Þ*t* is assumed to be an element of R. Resubstituting yields then the solution *I* and using *S* ¼ *w* � *I* yields *S*.

For more epidemic models on time scales that are systems of first order nonlinear dynamic equations, see [18–21]. While the dynamic Susceptible-Infected-Recovered epidemic model introduced in [18] can be solved explicitly via variable transformations, in most cases, including [19], explicit solutions to nonlinear dynamic equations are not available. In these cases, properties of solutions such as existence and uniqueness are of fundamental interest. The interested reader is referred to [22, Section 2] and [3, Section 8.2].

#### **Author details**

Sabrina Streipert McMaster University, Hamilton, Ontario, Canada

\*Address all correspondence to: streipes@mcmaster.ca

© 2022 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**

[1] May R. Simple mathematical models with very complicated dynamics. Nature. 1976;**261**:459-467

[2] Hilger S. Analysis on measure chains—A unified approach to continuous and discrete calculus. Research in Mathematics. 1990;**18**:18-56

[3] Bohner M, Peterson A. Dynamic Equations on Time Scales. Boston, MA: Birkhäuser Boston Inc.; 2001 ISBN 0-8176-4225-0. An introduction with applications

[4] Historical el nino/la nina episodes (1950–present), climate prediction center. Available from: https://origin. cpc.ncep.noaa.gov/products/analysis\_ monitoring/ensostuff/ONI\_v5.php [Accessed: March 15, 2019]

[5] Anderson DR, Krueger RJ, Peterson AC. Delay dynamic equations with stability. Advances in Difference Equations. 2006;**2006**:19

[6] Kelley WG, Peterson AC. Difference Equations: An Introduction with Applications. Boston, MA: Academic Press, Inc.; 1991 ISBN 0-12-403325-3

[7] Pötzsche C. Chain rule and invariance principle on measure chains. Journal of Computational and Applied Mathematics. 2002;**141**(1):249-254 ISSN 0377-0427. Dynamic Equations on Time Scales

[8] Keller S. Asymptotisches Verhalten invarianter Faserbundel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen. Augsburg: Universität Augsburg; 1999. Thesis (Ph.D.)

[9] Agarwal RP, Bohner M. Basic calculus on time scales and some of its

applications. Results in Mathematics. 1999;**35**(1–2):3-22

[10] Christiansen FB, Fenchel TM. Theories of Populations in Biological Communities. Ecological Studies, Berlin Heidelberg: Springer; 2012. Available from: https://books.google.de/books?id= HAL8CAAAQBAJ

[11] Tisdell CC, Zaidi A. Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. Nonlinear Analysis. 2008; **68**(11):3504-3524

[12] Bohner M, Warth H. The Beverton– Holt dynamic equation. Applicable Analysis. 2007;**86**(8):1007-1015

[13] Girejko E, Machado L, Malinowska AB, Martins N. On consensus in the Cucker-Smale type model on isolated time scales. Discrete & Continuous Dynamical System Series. 2018;**11**(1):77-89. ISSN 1937-1632

[14] Cucker F, Smale S. Emergent behavior in flocks. IEEE Transactions on Automatic Control. 2007;**52**(5): 852-862. DOI: 10.1109/TAC.2007. 895842

[15] Bohner M, Cuchta T, Streipert S. Delay dynamic equations on isolated time scales and the relevance of oneperiodic coefficients. Mathematicsl Methods in the Applied Sciences. 2022: 1-18. DOI: 10.1002/mma.8141

[16] Bohner M, Dannan FM, Streipert S. A nonautonomous Beverton-Holt equation of higher order. Journal of Mathematical Analysis and Applications. 2018;**457**(1):114-133

*Dynamic Equations on Time Scales DOI: http://dx.doi.org/10.5772/intechopen.104691*

[17] Bohner M, Streipert S. An integrable SIS model on time scales. In: Bohner M, Siegmund S, Šimon Hilscher R, Stehlík P, editors. Difference Equations and Discrete Dynamical Systems with Applications. Cham: Springer International Publishing; 2020. pp. 187-200

[18] Bohner M, Streipert S, Torres DFM. Exact solution to a dynamic sir model. Nonlinear Analysis Hybrid Systems. 2019;**32**:228-238

[19] Ferreira RAC, Silva CM. A nonautonomous epidemic model on time scales. Journal of Difference Equations and Applications. 2018;**24**(8):1295-1317

[20] Sae-Jie W, Bunwong K, Moore E. The effect of time scales on sis epidemic model. WSEAS Transactions on Mathematics. 2010;**9**(10):757-767

[21] Yeni G. Modeling of HIV, SIR and SIS Epidemics on Time Scales and Oscillation Theory. ProQuest LLC, Ann Arbor, MI: Missouri University of Science and Technology; 2019. ISBN 978-1392-67226-6. Thesis (Ph.D.)

[22] Lakshmikantham V, Kaymakçalan B, Sivasundaram S. Dynamic Systems on Measure Chains, Volume 370 of Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers; 1996

#### **Chapter 2**

## Existence Results for Boundary Value Problem of Nonlinear Fractional Differential Equation

*Noureddine Bouteraa and Habib Djourdem*

#### **Abstract**

In this chapter, we investigate the existence and uniqueness of solutions for class of nonlinear fractional differential equations with nonlocal boundary conditions. The existence results are obtained by using Leray-Schauder nonlinear alternative and Banach contraction principle. An illustrative example is presented at the end to illustrated the validity of our results.

**Keywords:** fractional differential equations, existence, nonlocal boundary, fixed-point theorem

#### **1. Introduction**

In this chapter, we are interested in the existence of solutions for nonlinear fractional difference equations

$$\mathbf{u}^{\varepsilon} \mathbf{D}\_{0^{+}}^{a} \mathbf{u}(\mathbf{t}) - \mathbf{A}^{\varepsilon} \mathbf{D}\_{0^{+}}^{\beta} \mathbf{u}(\mathbf{t}) = \mathbf{f}\left(\mathbf{t}, \mathbf{u}(\mathbf{t}), \,^{\varepsilon} \mathbf{D}\_{0^{+}}^{\beta} \mathbf{u}(\mathbf{t}), \,^{\varepsilon} \mathbf{D}\_{0^{+}}^{a} \mathbf{u}(\mathbf{t})\right), \quad \mathbf{t} \in \mathcal{J} = [\mathbf{0}, T], \tag{1}$$

subject to the three-point boundary conditions

$$\begin{cases} \lambda u(\mathbf{0}) - \mu u(T) - \gamma u(\eta) = d, \\ \lambda u'(\mathbf{0}) - \mu u'(T) - \gamma u'(\eta) = l, \end{cases} \tag{2}$$

where *<sup>T</sup>* <sup>&</sup>gt;0, 0<sup>≤</sup> *<sup>η</sup>*<sup>≤</sup> *<sup>T</sup>*, *<sup>λ</sup>* 6¼ *<sup>μ</sup>* <sup>þ</sup> *<sup>γ</sup>*, *<sup>d</sup>*, *<sup>l</sup>*, *<sup>λ</sup>*, *<sup>μ</sup>*, *<sup>γ</sup>* <sup>∈</sup> , *<sup>β</sup>* <sup>þ</sup> <sup>1</sup><sup>&</sup>lt; *<sup>α</sup>*, *<sup>A</sup>* is an *<sup>n</sup>*�*<sup>n</sup>* matrix and *<sup>c</sup> Dα* <sup>0</sup><sup>þ</sup> , *<sup>c</sup> Dβ* <sup>0</sup><sup>þ</sup> are the Caputo fractional derivatives of order 1<*α* ≤2, 0<*β* ≤ 1, respectively.

The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz and L'Hospital. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics,

electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electro-analytical chemistry, biology, control theory, fitting of experimental data, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details we refer the reader to [1–5] and the references cited therein.

Fractional differential equation theory have recieved increasing attention. This theory has been developed very quickly and attracted a considerable interest from researches in this field, which revealed the flexibility of fractional calculus theory in designing various mathematical models. The main methods conducted in this articles are by terms of fixed point techniques [6]. Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located, at intermediate points, see [7, 8] and the references therein. We quote also that realistic problems arising from economics, optimal control, stochastic analysis can be modeled as differential inclusion. The study of fractional differential inclusions was initiated by EL-Sayad and Ibrahim [9]. Also, recently, several qualitative results for fractional differential inclusion were obtained in [10–13] and the references therein.

The techniques of nonlinear analysis, as the main method to deal with the problems of nonlinear differential equations (DEs), nonlinear fractional differential equations (FDEs), nonlinear partial differential equations (PDEs), nonlinear fractional partial differential equations (FPDEs), nonlinear stochastic fractional partial differential equations (SFPDEs), plays an essential role in the research of this field, such as establishing the existence, uniqueness and multiplicity of solutions (or positive solutions) and mild solutions for nonlinear of different kinds of FPDEs, FPDEs, SFPDEs, inclusion differential equations and inclusion fractional differential equations with various boundary conditions, by using different techniques (approaches). For more details, see [14–36] and the references therein. For example, iterative method is an important tool for solving linear and nonlinear boundary value problems. It has been used in the research areas of mathematics and several branches of science and other fields. However, many authors showed the existence of positive solutions for a class of boundary value problem at resonance case. Some recent devolopment for resonant case can be found in [37, 38]. Let us cited few papers. In [39], the authors studied the boundary value problems of the fractional order differential equation:

$$\begin{cases} \begin{aligned} D\_{0+}^{a}\mathfrak{u}(t) = f(t, \mathfrak{u}(t)) = \mathbf{0}, \quad t \in (\mathbf{0}, 1), \\ \mathfrak{u}(\mathbf{0}) = \mathbf{0}, \; D\_{0+}^{\beta}\mathfrak{u}(\mathbf{1}) = aD\_{0+}^{\beta}\mathfrak{u}(\mathfrak{y}), \end{aligned} \end{cases}$$

where 1<*<sup>α</sup>* <sup>≤</sup>2, 0<*η*<1, 0 <sup>&</sup>lt;*a*, *<sup>β</sup>* <sup>&</sup>lt;1, *<sup>f</sup>* <sup>∈</sup>*<sup>C</sup>* ½ �� 0, 1 <sup>2</sup> , � � and *D<sup>α</sup>* <sup>0</sup>þ, *<sup>D</sup><sup>β</sup>* <sup>0</sup><sup>þ</sup> are the standard Riemann-Liouville fractional derivative of order *α*. They obtained the multiple positive solutions by the Leray-Schauder nonlinear alternative and the fixed point theorem on cones.

In 2017, Resapour et al. [40] investigated a Caputo fractional inclusion with integral boundary condition for the following problem

$$\begin{cases} \,^cD^\alpha u(t) \in F(t, u(t), ^cD^\beta u(t), u'(t)), \\ \quad u(0) + u'(0) + ^cD^\beta u(0) = \int\_0^\eta u(s)ds, \\ \quad u(1) + u'(1) + ^cD^\beta u(1) = \int\_0^\nu u(s)ds, \end{cases}$$

where 1<*<sup>α</sup>* <sup>≤</sup>2, *<sup>η</sup>*, *<sup>ν</sup>*, *<sup>β</sup>* <sup>∈</sup>ð Þ 0, 1 , *<sup>F</sup>* : ½ �� 0, 1 � � ! <sup>2</sup> is a compact valued multifunction and *<sup>c</sup> D<sup>α</sup>* denotes the Caputo fractional derivative of order *α*.

In 2017, Sheng and Jiang [41] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

$$\begin{cases} \ ^cD\_{0^+}^a u(t) - A^c D\_{0^+}^\beta u(t) = f(t, u(t)), \quad t \in [0, T], \\\ u(0) = u\_0, \qquad u'(0) = u'\_0, \end{cases}$$

where 0 <sup>&</sup>lt;*<sup>β</sup>* <sup>≤</sup>1<*<sup>α</sup>* <sup>≤</sup>2, 0<*<sup>T</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>u</sup>*<sup>∈</sup> *<sup>n</sup>*, *<sup>A</sup>* is an *<sup>n</sup>*�*<sup>n</sup>* matrix, *<sup>f</sup>* : ½ �� 0, 1 *<sup>n</sup>* ! *<sup>n</sup>* jointly continuous function and *<sup>c</sup> Dα* <sup>0</sup><sup>þ</sup> , *<sup>c</sup> Dβ* <sup>0</sup><sup>þ</sup> are the Caputo derivatives of order *α*, *β*, respectively.

In 2018, Abbes et al. [42] studied the existence and uniqueness of the solutions for fractional damped dynamical systems

$$\begin{cases} \ \ ^cD\_{0^+}^a u(t) - A ^cD\_{0^+}^\beta u(t) = f\left(t, u(t), \, ^cD\_{0^+}^\beta u(t), \, ^cD\_{0^+}^a u(t)\right), \quad t \in [0, T], \\\\ u(0) = u(T), \qquad u'(0) = u'(T), \end{cases}$$

where 0 <sup>&</sup>lt;*<sup>β</sup>* <sup>≤</sup>1<*<sup>α</sup>* <sup>≤</sup>2, 0<*<sup>T</sup>* <sup>&</sup>lt; <sup>∞</sup>, *<sup>u</sup>*<sup>∈</sup> *<sup>n</sup>*, *<sup>A</sup>* is an *<sup>n</sup>*�*<sup>n</sup>* matrix and *<sup>f</sup>* : ½ �� 0, 1 *<sup>n</sup>* ! *<sup>n</sup>* jointly continuous.

In 2019, Tao Zhu [43] studied the existence and uniqueness of positive solutions of the following fractional differential equations

$$\begin{cases} \quad D\_{0^+}^a u(t) - A^c D\_{0^+}^\beta u(t) = f(t, u(t)), \quad t \in [0, T), \ 0 < \beta < a < 1, \\\ u(0) = u\_0. \end{cases}$$

Inspired and motivated by the works mentioned above, we establish the existence results for the nonlocal boundary value problem (1.1)–(1.2) by using Leray-Schauder nonlinear alternative and the Banach fixed point theorem. Note that our work generalized the three works cited above [41–43]. The chapter is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. In Section 3, deals with main results and we give an example to illustrate our results.

#### **2. Existence and uniqueness results for our problem**

#### **2.1 Preliminaries**

Let as introduce notations, definitions and preliminary facts that will be need in the sequel. For more details, see for example [44–46].

Definition 2.1. The Caputo fractional derivative of order *α* for the function *<sup>u</sup>*∈*C<sup>n</sup>*ð Þ ½ Þ 0, <sup>∞</sup> , is defined by

$${}^{c}D\_{0^{+}}^{a}u(t) = \frac{1}{\Gamma(n-a)} \int\_{0}^{t} (t-s)^{n-a-1} u^{(n)}(s)ds.$$

where Γð Þ� is the Eleur gamma function and *α* >0, *n* ¼ ½ �þ *α* 1, ½ � *α* denotes the integer part of the real number *α*.

Definition 2.2. The Riemann-Liouville fractional integral of order *α*>0 of a function *u* : ð Þ! 0, ∞ is given by

$$I\_{0^{+}}^{a}u(t) = \frac{1}{\Gamma(a)} \int\_{0}^{t} (t-s)^{a-1} u(s)ds, \quad t > 0.$$

where Γð Þ� is the Eleur gamma function, provided that the right side is pointwise defined on 0, ð Þ ∞ .

Lemma 2.1. *Let u* <sup>∈</sup> *AC<sup>n</sup>*½ � 0, *<sup>T</sup>* , *<sup>n</sup>* <sup>∈</sup> *and u*ð Þ� <sup>∈</sup>*C*½ � 0, *<sup>T</sup> . Then, we have*

$${}^{c}D\_{0^{+}}^{\\\beta} \left(I\_{0^{+}}^{a}u(t)\right) = I\_{0^{+}}^{a-\beta}u(t),$$

$$I\_{0^{+}}^{a} \left({}^{c}D\_{0^{+}}^{a}u(t)\right) = u(t) - \sum\_{k=0}^{n-1} \frac{t^{k}}{k!} u^{(k)}(0), \ t>0, \ n-1$$

*Especially, when* 1<*α* <2*, then we have*

$$H\_{0^{+}}^{a} \left( {}^{c}D\_{0^{+}}^{a} \mu(t) \right) = \mu(t) - \mu(\mathbf{0}) - t\mu'(\mathbf{0}).$$

Lemma 2.2. 10 ð Þ ½ � *Let* 0<*β* <1<*α* <2*, then we have*

$$I\_{0^{+}}^{a}\left(^{c}D\_{0^{+}}^{\beta}\mathfrak{u}(t)\right) = I\_{0^{+}}^{a-\beta}\mathfrak{u}(t) - \frac{\mathfrak{u}(\mathbf{0})t^{a-\beta}}{\Gamma(a-\beta+1)}.$$

#### **2.2 Existence results**

Let *C J*, *<sup>n</sup>* ð Þ be the Banach space for all continuous function from *<sup>J</sup>* into *<sup>n</sup>* equipped with the norm

$$\|u\|\_{\ast} = \sup\{\|u(t)\|: t \in J\},$$

where k k� denotes a suitable complete norm on *<sup>n</sup>*. Denote *<sup>L</sup>*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ the Banach space of the measurable functions *<sup>u</sup>* : *<sup>J</sup>* ! *<sup>n</sup>* that are Lebesgue integrable with norm

$$\|\mathfrak{u}\|\_{L^1} = \int\_{\mathfrak{o}}^T \|\mathfrak{u}(t)\| dt.$$

Let *AC J*, *<sup>n</sup>* ð Þ be the Banach space of absolutely continuous valued functions on *<sup>J</sup>* and set

$$AC^n(f) = \left\{ u : f \to \mathbb{R}^n : u, u', u', \dots, u^{(n-1)} \in \mathbb{C}(f, \mathbb{R}^n) \right\} \text{ and } \ u^{(n-1)} \in AC(f, \mathbb{R}^n).$$

By

$$\mathbf{C}^1(f) = \{ \boldsymbol{\mu} : \boldsymbol{J} \to \mathbb{R}^n \text{ where } \boldsymbol{\mu}' \in \mathbf{C}(\boldsymbol{J}, \mathbb{R}^n) \},$$

we denote the Banach space equipped with the norm

$$||u||\_1 = \max\left\{||u||\_\infty, ||u'||\_\infty\right\}.$$

For the sake of brevity, we set

$$\begin{aligned} \delta &= (\lambda - \mu - \eta)\Gamma(a - \beta + 1) + A\left(\mu T^{a - \beta} + \eta \eta^{a - \beta}\right), \quad \Delta = \frac{\Gamma(a - \beta + 1)}{\delta} \\ \sigma &= A(a - \beta)\left(\mu T^{a - \beta - 1} + \eta \eta^{a - \beta - 1}\right), \quad \Lambda = (\lambda - \mu - \eta) - (\mu T + \eta \eta)\left(\frac{\sigma}{\delta}\right), \\ R\_1 &= \left(1 + \frac{||A||T^{a - \beta}}{\Gamma(a - \beta + 1)}\right)M\_1 + TM\_2 + \frac{||A||T^{a - \beta}}{\Gamma(a - \beta + 1)} \\ R\_2 &= M\_2 + \frac{(a - \beta)||A||T^{a - \beta - 1}}{\Gamma(a - \beta + 1)}M\_1 + \frac{||A||T^{a - \beta - 1}}{\Gamma(a - \beta)} \\ M\_1 &= \Delta\left\{\left[\Lambda^{-1}\left(\frac{\sigma}{\delta}\right)(\mu T + \eta \eta) + 1\right]\Phi + \Lambda^{-1}(\mu T + \eta \eta)\Theta\right\} \end{aligned}$$

,

and

$$M\_2 = \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \Phi + \Theta \right\} \,,$$

with

$$\begin{split} \Phi &= \frac{||A||}{\Gamma(a-\beta+1)} \left( \mu T^{a-\beta} + \eta \eta^{a-\beta} \right) + \frac{\left( \mu T^{a} + \eta \eta^{a} \right) \left( L\_{1} \Gamma(2-\beta) + T^{1-\beta} (L\_{3} ||A|| + L\_{2}) \right)}{\Gamma(a+1) \Gamma(2-\beta) (1-L\_{3})}, \\ \Theta &= \frac{||A||}{\Gamma(a-\beta)} \left( \mu T^{a-\beta-1} + \eta \eta^{a-\beta-1} \right) + \frac{\left( \mu T^{a-1} + \eta \eta^{a-1} \right) \left( L\_{1} \Gamma(2-\beta) + T^{1-\beta} (L\_{3} ||A|| + L\_{2}) \right)}{\Gamma(a) \Gamma(2-\beta) (1-L\_{3})}. \end{split}$$

Lemma 2.3. *Let y*ð Þ� <sup>∈</sup>*C J*, *<sup>n</sup>* ð Þ*. The function u*ð Þ� <sup>∈</sup>*C*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ *is a solution of the fractional differential problem*

$$\begin{cases} \,^cD\_{0^+}^a u(t) - A^c D\_{0^+}^\theta u(t) = \chi(t), \quad t \in J = [0, T], \\\quad \lambda u(0) - \mu u(T) - \gamma u(\eta) = d, \\\quad \lambda u'(0) - \mu u'(T) - \gamma u'(\eta) = l, \end{cases} \tag{3}$$

if and only if, *u* is a solution of the fractional integral equation

$$\begin{split} u(t) &= \left( 1 - \frac{At^{a-\beta}}{\Gamma(a-\beta+1)} \right) u(0) + tu'(0) + \frac{A}{\Gamma(a-\beta)} \int\_0^t (t-s)^{a-\beta-1} u(s) ds \\ &\quad + \frac{1}{\Gamma(a)} \int\_0^t (t-s)^{a-1} y(s) ds, \end{split} \tag{4}$$

with

$$\begin{split} u(\mathbf{0}) &= \Delta \left\{ \left[ \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \gamma \eta) + 1 \right] \left[ A I\_{0^+}^{\alpha - \beta} (\mu u(T) + \gamma u(\eta)) + I\_{0^+}^{\alpha} (\mu \eta(T) + \gamma \eta(\eta)) + d \right] \right\}, \\ &+ \Lambda^{-1} (\mu T + \gamma \eta) \left[ A I\_{0^+}^{\alpha - \beta - 1} (\mu u(T) + \gamma u(\eta)) + I\_{0^+}^{\alpha - 1} (\mu \eta(T) + \gamma \eta(\eta)) + l \right] \end{split} \tag{5}$$

and

$$\begin{split} u'(\mathbf{0}) &= \Lambda^{-1} \Big\{ \Big( \frac{\sigma}{\delta} \Big) \Big[ A I\_{0^{+}}^{a-\beta} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a} (\mu \jmath(T) + \gamma \jmath(\eta)) + d \Big] \\ &+ \Big[ A I\_{0^{+}}^{a-\beta-1} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a-1} (\mu \jmath(T) + \gamma \jmath(\eta)) \Big] + l \Big\}, \end{split} \tag{6}$$

where

$$\begin{aligned} I\_{0^+}^a u(T) &= \frac{1}{\Gamma(a)} \int\_0^T (T-s)^{a-1} u(s) ds, \\ I\_{0^+}^a u(\eta) &= \frac{1}{\Gamma(a)} \int\_0^\eta (\eta-s)^{a-1} u(s) ds, \\ I\_{0^+}^{a-\beta} u(T) &= \frac{1}{\Gamma(a-\beta)} \int\_0^T (T-s)^{a-\beta-1} u(s) ds, \\ I\_{0^+}^{a-\beta} u(\eta) &= \frac{1}{\Gamma(a-\beta)} \int\_0^\eta (\eta-s)^{a-\beta-1} u(s) ds. \end{aligned}$$

Proof. From Lemmas 2.1 and 2.2, we have

$$u(t) = u(0) + tu'(0) - \frac{At^{a-\beta}}{\Gamma(a-\beta+1)}u(0) + \frac{A}{\Gamma(a-\beta)}\int\_0^t (t-s)^{a-\beta-1}u(s)ds$$

$$+ \frac{1}{\Gamma(a)}\int\_0^t (t-s)^{a-1}\chi(s)ds$$

Applying conditions (2), we obtain (5) and (6).

Conversely, assume that *u* satisfies the fractional integral (4), and using the facts that *<sup>c</sup> Dα* <sup>0</sup><sup>þ</sup> is the left inverse of *I α* <sup>0</sup><sup>þ</sup> and the fact that *<sup>c</sup> Dα* <sup>0</sup><sup>þ</sup> *C* ¼ 0, where *C* is a constant, we get

$$f^\varepsilon D\_{0^+}^a u(t) - A^\varepsilon D\_{0^+}^\beta u(t) = f(t, u(t), u'(t)), \quad t \in I = [0, T].$$

Also, we can easily show that

$$\begin{cases} \lambda u(\mathbf{0}) - \mu u(T) - \eta u(\eta) = d, \\\lambda u'(\mathbf{0}) - \mu u'(T) - \eta u'(\eta) = l. \end{cases}$$

The proof is complete.

$$\left| I\_{0^{+}}^{a} \boldsymbol{\jmath}(\boldsymbol{\eta}) \right| = \left| \int\_{0}^{\eta} \frac{(\boldsymbol{\eta} - \boldsymbol{\tau})^{a-1}}{\Gamma(a)} \boldsymbol{\jmath}(\boldsymbol{\tau}) d\boldsymbol{\tau} \right| \leq \frac{\eta^{a}}{\Gamma(a+1)} \left\| \boldsymbol{\jmath} \right\|.$$

$$\int\_0^\eta \frac{(\eta - \tau)^{a-1}}{\Gamma(a)} \mathcal{Y}(\tau) d\tau = \left[ -\frac{(\eta - \tau)^a}{a\Gamma(a)} \right]\_0^\eta = \frac{\eta^a}{a\Gamma(a)} = \frac{s^a}{\Gamma(a+1)}.$$

$$\left| \int\_0^\eta \frac{(s-\tau)^{a-1}}{\Gamma(a)} \mathcal{Y}(\tau) d\tau \right| \le \frac{\eta^a}{\Gamma(a+1)} \|\mathcal{Y}\|\,.$$

$$\left\|{^cD\_{0^+}^\beta u(t)}\right\|\_{\infty} \le \frac{T^{1-\beta}}{\Gamma(2-\beta)} \left\|{u'}\right\|\_{\infty},$$

$$\left\|{^cD\_{0^+}^\beta u(t)}\right\|\_\infty \le \frac{T^{1-\beta}}{\Gamma(2-\beta)} \|u\|\_1.$$

$$\left|D\_{0^{+}}^{\beta}u(t)\right| = \frac{1}{\Gamma(1-\beta)}\left|\int\_{0}^{t}(t-s)^{-\beta}u'(s)ds\right|$$

$$\leq ||u'||\_{\infty}\frac{1}{\Gamma(1-\beta)}\int\_{0}^{t}(t-s)^{-\beta}ds$$

$$=||u'||\_{\infty}\frac{t^{1-\beta}}{\Gamma(1-\beta)}$$

$$\leq \frac{T^{1-\beta}}{\Gamma(1-\beta)}||u'||\_{\infty}$$

$$\leq \frac{T^{1-\beta}}{\Gamma(1-\beta)}||u'||\_{1}.$$

ð Þ *H*<sup>1</sup> there existe a constants *L*1, *L*<sup>2</sup> >0 and 0 <*L*<sup>3</sup> <1 such that

$$|f(t, u, v, w) - f(t, \overline{u}, \overline{v}, \overline{w})| \le L\_1 ||u - \overline{u}|| + L\_2 ||v - \overline{v}|| + L\_3 ||w - \overline{w}||,$$

for any *u*, *v*, *u*, *v*, *w* ∈ *<sup>n</sup>* and *t*∈*J*.

Now we are in a position to present the first main result of this paper. The existence results is based on Banach contraction principle.

Theorem 1.1. ([47] Banach's fixed point theorem). Let *C* be a non-empty closed subset of a Banach space *E*, then any contraction mapping *T* of *C* into itself has a unique fixed point.

Theorem 1.2. Assume that ð Þ *H*<sup>1</sup> holds. If

$$\max\left(R\_1, R\_2\right) < \mathbf{1},\tag{7}$$

then the boundary value problem (1.1)–(1.2) has a unique solution on *J*.

Proof. We transform the problem (1.1)–(1.2) into fixed point problem. Let *N* : *<sup>C</sup>*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ! *<sup>C</sup>*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ the operator defined by

$$\begin{split} \mathbf{1}(Nu)(t) &= \left(\mathbf{1} - \frac{At^{a-\beta}}{\Gamma(a-\beta+1)}\right) \mathbf{B} + t\mathbf{D} + \frac{A}{\Gamma(a-\beta)} \int\_{0}^{t} (t-s)^{a-\beta-1} u(s) ds \\ &+ \frac{1}{\Gamma(a)} \int\_{0}^{t} (t-s)^{a-1} \mathbf{g}(s) ds, \end{split} \tag{8}$$

with

$$\begin{split} B &= \Delta \left\{ \left[ \Lambda^{-1} \Big( \frac{\sigma}{\delta} \Big) (\mu T + \gamma \eta) + 1 \right] \Big[ A I\_{0^{+}}^{a-\beta} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a} (\mu \eta(T) + \gamma \eta(\eta)) + d \Big] \right\} \\ &+ \Lambda^{-1} (\mu T + \gamma \eta) \Big[ A I\_{0^{+}}^{a-\beta-1} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a-1} (\mu \eta(T) + \gamma \eta(\eta)) + l \Big] \Big), \end{split}$$

and

$$\begin{aligned} D &= \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \left[ A I\_{0^{+}}^{a-\beta} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a} (\mu \mathbf{g}(T) + \eta \mathbf{g}(\eta)) + d \right] \right\} \\ &+ \left[ A I\_{0^{+}}^{a-\beta-1} (\mu u(T) + \gamma u(\eta)) + I\_{0^{+}}^{a-1} (\mu \mathbf{g}(T) + \gamma \mathbf{g}(\eta)) \right] + l \right\}, \end{aligned}$$

where *<sup>g</sup>* <sup>∈</sup>*C J*, *<sup>n</sup>* ð Þ be such that

$$\mathbf{g}(t) = f\left(t, u(t), {}^{\varepsilon}D\_{0^{+}}^{\beta}u(t), \mathbf{g}(t) + A^{\varepsilon}D\_{0^{+}}^{\beta}u(t)\right),$$

For every *<sup>u</sup>*∈*C*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ and any *<sup>t</sup>*∈*J*, we have

$$\begin{split} f(N\boldsymbol{u})(t) &= D - \frac{(\boldsymbol{a} - \boldsymbol{\beta})A\boldsymbol{t}^{\alpha-\beta-1}}{\Gamma(\alpha-\beta+1)}\boldsymbol{B} + \frac{A}{\Gamma(\alpha-\beta-1)}\Big{\Big{}}\_{0}^{t}(t-s)^{\alpha-\beta-2}\boldsymbol{u}(s)ds \\ &+ \frac{1}{\Gamma(\alpha-1)}\int\_{0}^{t}(t-s)^{\alpha-2}\boldsymbol{g}(s)ds. \end{split} \tag{9}$$

**28**

Clearly, the fixed points of operator *N* are solutions of problem (1.1)–(1.2). It is clear that ð Þ *Nu* <sup>∈</sup>*C J*, *<sup>n</sup>* ð Þ, consequently, *<sup>N</sup>* is well defined. Let *<sup>u</sup>*, *<sup>v</sup>*∈*C J*, *<sup>n</sup>* ð Þ. Then for *<sup>t</sup>*∈*J*, we have

$$\begin{split} \| |(Nu)(t) - (N\nu)(t) | &\leq \left( 1 + \frac{||A||T^{a-\beta}}{\Gamma(a-\beta+1)} \right) \| B - B\_1 \| + T \| D - D\_1 \| \\ &+ \frac{||A||}{\Gamma(a-\beta)} \int\_0^T (T-s)^{a-\beta-1} |u(s) - v(s)| |ds \\ &+ \frac{1}{\Gamma(a)} \int\_0^T (T-s)^{a-1} |g(s) - h(s)| |ds, \end{split}$$

with

$$\begin{split} B\_{1} &= \Delta \left\{ \left[ \Lambda^{-1} \binom{\sigma}{\delta} (\mu T + \eta \eta) + \mathbf{1} \right] \left[ A I\_{0^{+}}^{a-\beta} (\mu \nu(T) + \eta \nu(\eta)) + I\_{0^{+}}^{a} (\mu h(T) + \eta h(\eta)) + d \right] \right\} \\ &+ \Lambda^{-1} (\mu T + \eta \eta) \left[ A I\_{0^{+}}^{a-\beta-1} (\mu \nu(T) + \eta \nu(\eta)) + I\_{0^{+}}^{a-1} (\mu h(T) + \eta h(\eta)) + l \right] \}, \end{split}$$

and

$$\begin{aligned} D\_1 &= \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \left[ A I\_{0^+}^{a-\beta} (\mu v(T) + \eta v(\eta)) + I\_{0^+}^a (\mu h(T) + \eta h(\eta)) + d \right] \right\} \\ &+ \left[ A I\_{0^+}^{a-\beta-1} (\mu v(T) + \eta v(\eta)) + I\_{0^+}^{a-1} (\mu h(T) + \eta h(\eta)) \right] + l \right\}, \end{aligned}$$

From ð Þ *H* , for any *t*∈*J*, we have

$$\begin{split} \left\lVert \mathbf{g}(t) - h(t) \right\rVert &= L\_{1} \lVert u(t) - v(t) \rVert + L\_{2} \left\lVert \begin{aligned} ^{c}D\_{0^{+}}^{\theta}u(t) - ^{c}D\_{0^{+}}^{\theta}v(t) \right\rVert \\ + L\_{3} \left\lVert \mathbf{g}(t) + A^{c}D\_{0^{+}}^{\theta}u(t) - h(t) - A^{c}D\_{0^{+}}^{\theta}v(t) \right\rVert \\ \leq L\_{1} \lVert u(t) - v(t) \rVert + L\_{2} \left\lVert ^{c}D\_{0^{+}}^{\theta}u(t) - ^{c}D\_{0^{+}}^{\theta}v(t) \right\rVert \\ &+ L\_{3} \lVert \mathbf{g}(t) - h(t) \rVert + L\_{3} \lVert A \rVert \left\lVert ^{c}D\_{0^{+}}^{\theta}u(t) - ^{c}D\_{0^{+}}^{\theta}v(t) \right\rVert \\ \leq L\_{1} \lVert u(t) - v(t) \rVert + L\_{3} \lVert \mathbf{g}(t) - h(t) \rVert + (L\_{3} \lVert A \rVert + L\_{2}) \left\lVert ^{c}D\_{0^{+}}^{\theta}(u(t) - v(t)) \right\rVert \end{split} $$

Thus

$$\begin{split} \left\| \left\| \mathbf{g}(t) - h(t) \right\| \right\| &\leq \frac{L\_1}{\mathbf{1} - L\_3} \left\| u(t) - v(t) \right\| + \frac{L\_3 \left\| A \right\| + L\_2}{\mathbf{1} - L\_3} \left\| \left\| D\_{\mathbf{0}^+}^\beta (u(t) - v(t)) \right\| \right\| \\ &\leq \frac{L\_1}{\mathbf{1} - L\_3} \left\| u - v \right\|\_\infty + \frac{L\_3 \left\| A \right\| + L\_2}{\mathbf{1} - L\_3} \left\| \left\| D\_{\mathbf{0}^+}^\beta (u - v) \right\| \right\|\_\infty. \end{split}$$

Then, according to the Lemma 3.2, we get

$$\begin{split} \|\mathbf{g}(t) - h(t)\| &\leq \frac{L\_1}{\mathbf{1} - L\_3} \|\boldsymbol{u} - \boldsymbol{v}\|\_1 + \frac{T^{1-\beta}(L\_3 \|\boldsymbol{A}\| + L\_2)}{\Gamma(2-\beta)(\mathbf{1} - L\_3)} \|\boldsymbol{u} - \boldsymbol{v}\|\_1 \\ &= \frac{L\_1 \Gamma(2-\beta) + T^{1-\beta}(L\_3 \|\boldsymbol{A}\| + L\_2)}{\Gamma(2-\beta)(\mathbf{1} - L\_3)} \|\boldsymbol{u} - \boldsymbol{v}\|\_1. \end{split} \tag{10}$$

By employing (10) and Lemma 3.1, we get

$$\begin{split} \| |B\_1 - B\_2| \| \le & \Delta \left\{ \left[ \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \gamma \eta) + 1 \right] \Phi + \Lambda^{-1} (\mu T + \gamma \eta) \Theta \right\} || u - v||\_1 \\ = & \mathcal{M}\_1 || u - v||\_1. \end{split}$$

and

$$\begin{aligned} ||D\_1 - D\_2|| \leq \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \Phi + \Theta \right\} ||u - v||\_1 \\ &= M\_2 ||u - v||\_1, \end{aligned}$$

where Φ and Θ defined above. Thus, for *t*∈*J*, we have

$$\begin{split} \| (\mathcal{N}u)(t) - (\mathcal{N}v)(t) \| &\leq \left[ \left( 1 + \frac{||A||T^{\alpha-\beta}}{\Gamma(\alpha-\beta+1)} \right) M\_1 + T\mathcal{M}\_2 + \frac{||A||T^{\alpha-\beta}}{\Gamma(\alpha-\beta+1)} \right] \\ &+ \frac{T^{\alpha}L\_1\Gamma(2-\beta) + T^{1-\beta+a}(L\_3||A||+L\_2)}{\Gamma(a+1)\Gamma(2-\beta)(1-L\_2)} \| |u-v| \| \_1 \\ &= R\_1 \| u - v \| \_1. \end{split}$$

Also

$$\begin{split} \| |(Nu)(t) - (N\nu)(t) | &\leq \| D\_2 - D\_1 \| \| + \frac{(a-\beta) \| A \| \| T^{\alpha - \beta - 1} \|}{\Gamma(a - \beta + 1)} \| B\_1 - B\_2 \| \\ &\quad + \frac{\| A \|}{\Gamma(a - \beta - 1)} \int\_0^T (T - s)^{a - \beta - 2} \| u(s) - v(s) \| ds \\ &\quad + \frac{1}{\Gamma(a - 1)} \int\_0^T (T - s)^{a - 2} \| g(s) - h(s) \| ds. \end{split}$$

By employing (10) and Lemma 3.2, we get

$$\begin{aligned} \| (Nu)(t) - (N\nu)(t) \| &\le \left[ M\_2 + \frac{(a-\beta) \| A \| \| T^{a-\beta-1} \mathbf{1} \|\_{1}}{\Gamma(a-\beta+\mathbf{1})} M\_1 + \frac{\| A \| \| T^{a-\beta-1} \mathbf{1} \|}{\Gamma(a-\beta)} \right. \\ &+ \frac{T^{a-1} L\_1 \Gamma(2-\beta) + T^{a-\beta} (L\_3 \| A \| \| + L\_2)}{\Gamma(a) \Gamma(2-\beta) (1-L\_2)} \| \| u - v \| \|\_{1} \\ &= R\_2 \| u - v \| \_1. \end{aligned}$$

Therefore

$$||(\mathcal{N}\boldsymbol{\mu})(t) - (\mathcal{N}\boldsymbol{\nu})(t)|| \le \max\left\{R\_1, R\_2\right\}||\boldsymbol{\mu} - \boldsymbol{\nu}||\_1.$$

Thus, by (10) the operator *N* is a contraction. Hence it follows by Banach's contraction principle that the boundary value problem (1)–(12) has a unique solution on *J*.

Now we are in a position to present the second main result of this paper. The existence results is based on Leray-Schauder nonlinear alternative.

Theorem 1.3. ([6] Nonlinear alternative for single valued maps). Let *E* be a Banach space, *C* a closed, convex subset of *E* and *U* an open subset of *C* with 0∈ *U*. Suppose that *F* : *U* ! *C* is a continuous and compact (that is *F U* � � is relatively compact subset of *C*) map. Then either

i. *F* has a fixed point in *U*, or

ii. there is a *<sup>u</sup>*∈*∂<sup>U</sup>* (the boundary of *<sup>U</sup>* in *<sup>C</sup>*) and *<sup>λ</sup>*<sup>∈</sup> ð Þ 0, 1 with *<sup>u</sup>* <sup>¼</sup> *<sup>λ</sup>F u*ð Þ.

Set

$$M\_1 = M\_3 + TM\_4 + \frac{||A||T^{a-\beta}}{\Gamma(a-\beta+1)}M\_3 + TM\_4 + \frac{||A||rT^{a-\beta}}{\Gamma(a-\beta+1)} + \frac{T^a}{\Gamma(a+1)}M\_4,$$

and

$$l\_2 = M\_4 + \frac{(\alpha - \beta) ||A||T^{\alpha - \beta - 1}}{\Gamma(a - \beta + 1)} M\_3 + \frac{||A||rT^{\alpha - \beta - 1}}{\Gamma(a - \beta)} + \frac{T^a M}{\Gamma(a)}.$$

Theorem 1.4. Assume that ð Þ *H*<sup>1</sup> holds and there exists a positive constant *M* >0 such that max f g¼ *l*1, *l*<sup>2</sup> *l* < *M*. Then the boundary value problem (1.1)–(1.2) has at least one solution on *J*.

Proof. Let *N* be the operator defined in (8).

*<sup>N</sup>* is continuous. Let ð Þ *un* be a sequence such that *un* ! *<sup>u</sup>* in *C J*, *<sup>n</sup>* ð Þ. Then for *<sup>t</sup>*<sup>∈</sup> *<sup>J</sup>*, we have

$$\begin{split} \| |(Nu)(t) - (Nu\_n)(t) | &\leq \left( 1 + \frac{||A||T^{a-\beta}}{\Gamma(a-\beta+1)} \right) \| |B\_1 - B\_{n2}| | + T \| D\_1 - D\_{n2} \| \\ &\quad + \frac{||A||}{\Gamma(a-\beta)} \int\_0^T (T-s)^{a-\beta-1} ||u(s) - u\_n(s)| |ds \\ &\quad + \frac{1}{\Gamma(a)} \int\_0^T (T-s)^{a-1} ||g(s) - g\_n(s)| |ds, \end{split}$$

where *Bn*2, *Dn*<sup>2</sup> ∈ *<sup>n</sup>*, with

$$\begin{split} B\_{n2} &= \Delta \left\{ \left[ \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \eta \eta) + 1 \right] \left[ A I\_{0^{+}}^{a-\beta} (\mu u\_{\pi}(T) + \gamma u\_{\pi}(\eta)) + I\_{0^{+}}^{a} (\mu \mathbf{g}\_{\pi}(T) + \eta \mathbf{g}\_{\pi}(\eta)) + d \right] \right. \\ &\left. + \Lambda^{-1} (\mu T + \eta \eta) \left[ A I\_{0^{+}}^{a-\beta-1} (\mu u\_{\pi}(T) + \gamma u\_{\pi}(\eta)) + I\_{0^{+}}^{a-1} (\mu \mathbf{g}\_{\pi}(T) + \eta \mathbf{g}\_{\pi}(\eta)) + l \right] \right\}, \\ D\_{n2} &= \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \left[ A I\_{0^{+}}^{a-\beta} (\mu u\_{\pi}(T) + \gamma u\_{\pi}(\eta)) + I\_{0^{+}}^{a} \left( \mu \mathbf{g}\_{\pi}(T) + \eta \mathbf{g}\_{\pi}(\eta) \right) + d \right] \right. \\ &\left. + A I\_{0^{+}}^{a-\beta-1} (\mu u\_{\pi}(T) + \gamma u\_{\pi}(\eta)) + I\_{0^{+}}^{a-1} \left( \mu \mathbf{g}\_{\pi}(T) + \gamma \mathbf{g}\_{\pi}(\eta) \right) + l \right\}, \end{split}$$

and

$$\mathbf{g}\_n(t) = f\left(t, u\_n(t), \,^\varepsilon D\_{0^+}^\beta u\_n(t), \mathbf{g}\_n(t) + A^\varepsilon D\_{0^+}^\beta u\_n(t)\right).$$

$$\left\|\mathbf{g}(t) - \mathbf{g}\_n(t)\right\| \le \frac{L\_1}{\mathbf{1} - L\_3} \left\|\boldsymbol{\mu}(t) - \boldsymbol{\mu}\_n(t)\right\| + \frac{L\_3 \left\|\boldsymbol{A}\right\| + L\_2}{\mathbf{1} - L\_3} \left\|\boldsymbol{\epsilon}\boldsymbol{D}\_{0^+}^\beta \left(\boldsymbol{u}(t) - \boldsymbol{u}\_n(t)\right)\right\|$$

$$\le \frac{L\_1}{\mathbf{1} - L\_3} \left\|\boldsymbol{u} - \boldsymbol{u}\_n\right\|\_\infty + \frac{L\_3 \left\|\boldsymbol{A}\right\| + L\_2}{\mathbf{1} - L\_3} \left\|\boldsymbol{\epsilon}\boldsymbol{D}\_{0^+}^\beta \left(\boldsymbol{u} - \boldsymbol{u}\_n\right)\right\|\_\infty.$$

$$\left||\mathbf{g}(t) - \mathbf{g}\_n(t)|| \le \frac{L\_1}{\mathbf{1} - L\_3} ||u - u\_n||\_1 + \frac{T^{1-\beta}(L\_3||A|| + L\_2)}{(\mathbf{1} - L\_3)\Gamma(2-\beta)} ||u - u\_n||\_1$$

$$= \frac{L\_1\Gamma(2-\beta) + T^{1-\beta}(L\_3||A|| + L\_2)}{(\mathbf{1} - L\_3)\Gamma(2-\beta)} ||u - u\_n||\_1.$$

$$\begin{aligned} ||B\_1 - B\_{n2}|| &\leq \Delta \left\{ \left[ \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \gamma \eta) + \mathbf{1} \right] \Phi + \Lambda^{-1} (\mu T + \gamma \eta) \Theta \right\} ||u - u\_n||\_1, \\ &= M\_1 ||u - u\_n||\_1. \end{aligned}$$

$$\begin{aligned} \| |D\_1 - D\_{n2}| \| \le \Lambda^{-1} \left\{ \left( \frac{\sigma}{\delta} \right) \Phi + \Theta \right\} \| u - u\_n \| \|\_{1} \\ &= M\_2 \| u - u\_n \| \_1, \end{aligned}$$

$$\begin{split} \| (Nu)(t) - (Nu\_n)(t) \| &\leq \left[ \left( 1 + \frac{||A||T^{\alpha-\beta}}{\Gamma(\alpha-\beta+1)} \right) \mathcal{M}\_1 + T\mathcal{M}\_2 + \frac{||A||T^{\alpha-\beta}}{\Gamma(\alpha-\beta+1)} \right. \\ &\quad + \frac{T^{\alpha}L\_1\Gamma(2-\beta) + T^{1-\beta+a}(L\_3||A||+L\_2)}{\Gamma(a+1)\Gamma(2-\beta)(1-L\_2)} \| |u-u\_n| \|\_1 \\ &= R\_1 ||u-u\_n||\_1. \end{split}$$

Also

$$\begin{split} \| (Nu)(t) - (Nu\_n)(t) \| &\le \| D\_{n2} - D\_1 \| + \frac{(\alpha - \beta) \| A \| \| T^{\alpha - \beta - 1} \|}{\Gamma(\alpha - \beta + 1)} \| B\_1 - B\_{n2} \| \\ &+ \frac{\| |A| \|}{\Gamma(\alpha - \beta - 1)} \int\_0^T (T - s)^{\alpha - \beta - 2} \| u(s) - u\_n(s) \| ds \\ &+ \frac{1}{\Gamma(\alpha - 1)} \int\_0^T (T - s)^{\alpha - 2} \| g(s) - g\_n(s) \| \| ds. \end{split}$$

By employing (10), we get

$$\begin{split} \left| ||(Nu)(t) - (Nu\_{\boldsymbol{\alpha}})(t)|| \right| &\leq \left[ M\_{2} + \frac{(\boldsymbol{\alpha} - \boldsymbol{\beta}) ||A||T^{\alpha - \beta - 1}}{\Gamma(\alpha - \beta + 1)} M\_{1} + \frac{||A||T^{\alpha - \beta - 1}}{\Gamma(\alpha - \beta)} \right. \\ &+ \frac{T^{\alpha - 1} L\_{1} \Gamma(2 - \beta) (L\_{3} ||A|| + L\_{2}) T^{\alpha - \beta - 1}}{\Gamma(\alpha - \beta + 1)} M\_{1} + \frac{||A||T^{\alpha - \beta - 1}}{\Gamma(\alpha - \beta)} \left|| u - u\_{\boldsymbol{\alpha}} \right||\_{1} . \end{split}$$

Thus k k *Nu* � *Nun* <sup>1</sup> ! 0 as *n* ! ∞, which implies that the operator *N* is continuous.

Now, we show *<sup>N</sup>* maps bounded sets into bounded sets in *C J*, *<sup>n</sup>* ð Þ. For a positive number *<sup>r</sup>*, let *Br* <sup>¼</sup> *<sup>u</sup>* <sup>∈</sup>*C*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ : k k*<sup>u</sup>* <sup>1</sup> <sup>≤</sup>*<sup>r</sup>* � � be a bounded set in *C J*, *<sup>n</sup>* ð Þ. Then we have

$$\begin{aligned} \|\mathbf{g}(t)\| &\le \left\| f\left(t, u(t), \mathbf{g}(t) + A^c D\_{0^+}^\theta u(t), D\_{0^+}^\theta u(t) \right) - f(t, \mathbf{0}, \mathbf{0}, \mathbf{0}) \right\| + \| f(t, \mathbf{0}, \mathbf{0}, \mathbf{0}) \| \\\\ &\le L\_1 \|u(t)\| + L\_3 \left\| \mathbf{g}(t) + A^c D\_{0^+}^\theta u(t) \right\| + L\_2 \left\| D\_{0^+}^\theta u(t) \right\| + \| f(t, \mathbf{0}, \mathbf{0}, \mathbf{0}) \| \\\\ &\le L\_1 \|u\|\_{\infty} + L\_3 \|\mathbf{g}(t)\| + (L\_3 \|A\| + L\_2) \left\| D\_{0^+}^\theta u \right\|\_{\infty} + f^\*, \end{aligned}$$

where sup *t*∈*J* j j *f t*ð Þ , 0, 0, 0 <sup>¼</sup> *<sup>f</sup>* <sup>∗</sup> <sup>&</sup>lt; <sup>∞</sup>. Thus

$$||\mathbf{g}(t)|| \le \frac{L\_1}{\mathbf{1} - L\_3} ||u||\_\circ + \frac{L\_3 ||A|| + L\_2}{\mathbf{1} - L\_3} \left|| D\_{0^+}^\beta u \right||\_\circ + \frac{f^\*}{\mathbf{1} - L\_3}.$$

Then, By Lemma 3.2, we have

$$\|\|g(t)\|\| \le \frac{L\_1}{1 - L\_3} \|u\|\|\_{\infty} + \frac{(L\_3 \|A\| + L\_2)T^{1-\beta}}{(1 - L\_3)\Gamma(2 - \beta)} \|u'\|\|\_{\infty} + \frac{f^\*}{1 - L\_3}$$

$$\le \frac{L\_1}{1 - L\_3} \|u\|\_1 + \frac{(L\_3 \|A\| + L\_2)T^{1-\beta}}{(1 - L\_3)\Gamma(2 - \beta)} \|u\|\_1 + \frac{f^\*}{1 - L\_3} \tag{11}$$

$$\le \frac{L\_1 r}{1 - L\_3} + \frac{(L\_3 \|A\| + L\_2)rT^{1-\beta}}{(1 - L\_3)\Gamma(2 - \beta)} + \frac{f^\*}{1 - L\_3} = M,$$

$$\begin{split} ||B|| &\le r||A||\Delta \left[ \left( \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \eta \eta) + 1 \right) \left( \frac{\mu T^{a-\beta} + \eta \eta^{a-\beta}}{\Gamma(a-\beta+1)} \right) \\ &+ \Lambda^{-1} (\mu T + \eta \eta) \left( \frac{\mu T^{a-\beta-1} + \eta \eta^{a-\beta-1}}{\Gamma(a-\beta)} \right) \right] \\ &+ M\Delta \left[ \left( \Lambda^{-1} \left( \frac{\sigma}{\delta} \right) (\mu T + \eta \eta) + 1 \right) \left( \frac{\mu T^a + \eta \eta^a}{\Gamma(a+1)} \right) + \Lambda^{-1} (\mu T + \eta \eta) \left( \frac{\mu T^{a-1} + \eta \eta^{a-1}}{\Gamma(a)} \right) \right] \\ &+ \Delta \Lambda^{-1} (\mu T + \eta \eta) \left[ 1 + d \left( \left( \frac{\sigma}{\delta} \right) + 1 \right) \right] = M\_3, \end{split}$$

$$||D|| \le r||A||\left[\Lambda^{-1}\left(\frac{\sigma}{\delta}\right)\frac{\mu T^{a-\beta} + \gamma \eta^{a-\beta}}{\Gamma(a-\beta+1)} + \frac{\mu T^{a-\beta-1} + \gamma \eta^{a-\beta-1}}{\Gamma(a-\beta)}\right]$$

$$+ \mathcal{M}\Lambda^{-1}\left[\left(\frac{\sigma}{\delta}\right)\left(\frac{\mu T^a + \gamma \eta^a}{\Gamma(a+1)}\right) + \left(\frac{\mu T^{a-1} + \gamma \eta^{a-1}}{\Gamma(a)}\right)\right] + \Lambda^{-1}\left[\left(\frac{\sigma}{\delta}\right)d + l\right] = \mathcal{M}\_4\Lambda^{-1}\left[\frac{\eta^{a-\beta}}{\delta}\right] + \left(\frac{\mu T^a + \gamma \eta^{a-\beta}}{\delta}\right)\left(\frac{\mu T^a + \gamma \eta^{a-\beta}}{\delta}\right)$$

$$\|(Nu)(t)\| \le M\_3 + \frac{||A||T^{\alpha-\beta}}{\Gamma(a-\beta+1)}M\_3 + TM\_4 + \frac{||A||rT^{\alpha-\beta}}{\Gamma(a-\beta+1)} + \frac{T^{\alpha}}{\Gamma(a+1)}M = l\_1,$$

$$\|(Nu)(t)\| \le M\_4 + \frac{(a-\beta)\|A\|\|T^{a-\beta-1}}{\Gamma(a-\beta+1)}M\_3 + \frac{||A||rT^{a-\beta-1}}{\Gamma(a-\beta)} + \frac{T^aM}{\Gamma(a)} = l\_2.$$

$$\|(Nu)\|\_1 \le \max\left\{l\_1, l\_2\right\} = l.\tag{12}$$

$$\begin{aligned} &\|(Nu)(t\_2) - (Nu)(t\_1)\| \leq M\_4(t\_2 - t\_1) + \left(1 + \frac{\|A\|\|M\_3}{\Gamma(\alpha - \beta + 1)}\right) \left(t\_2^{a-\beta} - t\_1^{a-\beta}\right) \\ &+ \frac{\|A\|\|r\|}{\Gamma(\alpha - \beta)} \int\_{t\_1}^{t\_2} (t\_2 - s)^{a-\beta-1} ds + \frac{\|A\|\|r\|}{\Gamma(\alpha - \beta)} \left[\left(t\_2 - s\right)^{a-\beta-1} - (t\_1 - s)^{a-\beta-1}\right] ds \\ &+ \frac{M\_1}{\Gamma(\alpha)} \left[\int\_{t\_1}^{t\_2} (t\_2 - s)^{a-1} ds + \int\_0^{t\_1} \left[\left(t\_2 - s\right)^{a-1} - (t\_1 - s)^{a-1}\right] ds\right] \end{aligned}$$

$$\begin{split} \left| \left( (Nu)(t\_2) - (Nu)(t\_1) \right) \right| &\leq \frac{(\alpha - \beta) \| A \| M\_3}{\Gamma(\alpha - \beta + 1)} \left( t\_2^{\alpha - \beta - 1} - t\_1^{\alpha - \beta - 1} \right) \\ &+ \frac{\| A \| r}{\Gamma(\alpha - \beta - 1)} \int\_{t\_1}^{t\_2} (t\_2 - s)^{\alpha - \beta - 2} ds + \frac{\| A \| r}{\Gamma(\alpha - \beta - 2)} \int\_0^{t\_1} [(t\_2 - s)^{\alpha - \beta - 2} - (t\_1 - s)^{\alpha - \beta - 2}] ds \\ &+ \frac{M}{\Gamma(\alpha - 1)} \left[ \int\_{t\_1}^{t\_2} (t\_2 - s)^{\alpha - 2} ds + \int\_{t\_1}^{t\_1} [(t\_2 - s)^{\alpha - 2} - (t\_1 - s)^{\alpha - 2}] ds \right] \end{split}$$

Again, it is seen that the right-hand side of the above inequality tends to zero as *t*<sup>2</sup> ! *t*1. Thus, k k ð Þ *Nu* ð Þ� *t*<sup>2</sup> ð Þ *Nu* ð Þ *t*<sup>1</sup> ! 0, as *t*<sup>2</sup> ! *t*1. This shows that the operator *N* is completely continuous, by the Ascoli-Arzela theorem. Thus, the operator *N* satisfies all the conditions of Theorem 3.4, and hence by its conclusion, either condition (i) or condition (ii) holds. We show that the condition (ii) is not possible.

Let *<sup>U</sup>* <sup>¼</sup> *<sup>u</sup>*∈*C*<sup>1</sup> *<sup>J</sup>*, *<sup>n</sup>* ð Þ : k k*<sup>u</sup>* <sup>&</sup>lt; *<sup>M</sup>* � � with max f g *<sup>l</sup>*1, *<sup>l</sup>*<sup>2</sup> <sup>¼</sup> *<sup>l</sup>*<sup>&</sup>lt; *<sup>M</sup>*. In view of condition *l* < *M* and by (12), we have

$$||Nu|| \le \max\left\{l\_1, l\_2\right\} < M.$$

Now, suppose there exists *<sup>u</sup>*<sup>∈</sup> *<sup>∂</sup><sup>U</sup>* and *<sup>λ</sup>*<sup>∈</sup> ð Þ 0, 1 such that *<sup>u</sup>* <sup>¼</sup> *<sup>λ</sup>Nu*. Then for such a choice of *u* and the constant *λ*, we have

$$M = ||u|| = \lambda ||\mathbf{N}u|| < \max\left\{l\_1, l\_2\right\} < M,$$

which is a contradiction. Consequently, by the Leray-Schauder alternative, we deduce that *F* has a fixed point *u* ∈ *U* which is a solution of the boundary value problem (1)–(12). The proof is completed.

We construct an example to illustrate the applicability of the results presented. Example 2.1. *Consider the following fractional differential equation*

$$\mathbf{u}^{\varepsilon}D\_{0^{+}}^{a}u(t) - A^{\varepsilon}D\_{0^{+}}^{\beta}u(t) = f\left(t, u(t), {}^{\varepsilon}D\_{0^{+}}^{\beta}, u(t), {}^{\varepsilon}D\_{0^{+}}^{a}u(t)\right), \quad t \in J = [0, 1], \tag{13}$$

subject to the three-point boundary conditions

$$\begin{cases} \ u(\mathbf{0}) - u(\mathbf{1}) - u\left(\frac{\mathbf{1}}{2}\right) = \mathbf{1}, \\\\ u'(\mathbf{0}) - u'(\mathbf{1}) - u'\left(\frac{\mathbf{1}}{2}\right) = \mathbf{1}, \end{cases} \tag{14}$$

where *<sup>α</sup>* <sup>¼</sup> 2, *<sup>β</sup>* <sup>¼</sup> 1, *<sup>λ</sup>* <sup>¼</sup> *<sup>μ</sup>* <sup>¼</sup> *<sup>d</sup>* <sup>¼</sup> *<sup>l</sup>* <sup>¼</sup> 1, *<sup>A</sup>* <sup>¼</sup> 2 1 0 2 � � and

$$\,\_2f\_i(t,u,v,w) = \frac{c\_it}{8} \arctan\left(|u| + |v| + |w|\right), \; i = 1, \; 2, \; v$$

such that *f* ¼ *f* <sup>1</sup>, *f* <sup>2</sup> � � with 0<*ci* <sup>&</sup>lt;1, *<sup>i</sup>* <sup>¼</sup> 1, 2. *For every ui*, *vi* ∈ <sup>2</sup> , *i* ¼ 1, 2, 3*, we have*

$$\left| \int f\_i(t, u\_1, u\_2, u\_3) - f\_j(t, v\_1, v\_2, v\_3) \right| \le \frac{c\_i}{8} \left( |u\_1 - v\_1| + |u\_2 - v\_2| + |u\_3 - v\_3| \right), i = 1, 2, 3$$

where *<sup>L</sup>*<sup>1</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> <sup>¼</sup> *<sup>L</sup>*<sup>3</sup> <sup>¼</sup> *ci* <sup>8</sup> for appropriate choice of constants *ci*, *i* ¼ 1, 2. we check the condition of Theorem 2.2. Clearly, assumption ð Þ *H*<sup>1</sup> holds. A simple computations of *R*1, *R*2, *l*<sup>1</sup> and *l*<sup>2</sup> shows tha the second condition of Theorems 3.3 and 3.5 is satisfied. Thus the conclusion of Theorems 3.3 and 3.5 applies, and hence the problem (13)–(14) has a unique solution and at least one solution on 0, 1 ½ �.

#### **3. Conclusions**

This chapter concerns the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal boundary conditions. By using Leray-Schauder nonlinear alternative and the Banach fixed point theorem, we shows the existence and uniqueness of positive solutions of our problem. In addition, an example is provided to demonstrate the effectiveness of the main results. The results of the present chapter are significantly contribute to the existing literature on the topic.

#### **Acknowledgements**

The authors want to thank the anonymous referee for the thorough reading of the manuscript and several suggestions that help us improve the presentation of the chapter.

#### **Conflict of interest**

The authors declare no conflict of interest.

### **Author details**

Noureddine Bouteraa1,2\*† and Habib Djourdem1,3\*†

1 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran, Ahmed Benbella, Algeria

2 Oran Graduate School of Economics, Bir El Djir, Algeria

3 University of Ahmed Zabbana, Relizane, Algeria

\*Address all correspondence to: bouteraa-27@hotmail.fr and djourdem.habib7@gmail.com

† These authors contributed equally.

© 2022 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**

[1] Kac V, Cheung P. Quantum Calculus. New York: Springer; 2002

[2] Lakshmikantham V, Vatsala AS. General uniqueness and monotone iterative technique for fractional differential equations. Applied Mathematics Letters. 2008;**21**(8): 828-834

[3] Miller S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: John Wiley and Sons, Inc.; 1993

[4] Rudin W. Functional analysis. In: International Series in Pure and Applied Mathematics. 2nd ed. New York: McGraw Hill; 1991

[5] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. Yverdon: Gordon & Breach; 1993

[6] Deimling K. Functional Analysis. Berlin: Springer; 1985

[7] Jarad F, Abdeljaw T, Baleanu D. On the generalized fractional derivatiives and their Caputo modification. Journal of Nonlinear Sciences and Applications. 2017;**10**(5):2607-2619

[8] Tian Y. Positive solutions to m-point boundary value problem of fractional differential equation. Acta Mathematicae Applicatae Sinica, English Series. 2013; **29**:661-672

[9] EL-Sayed AMA, EL-Haddad FM. Existence of integrable solutions for a functional integral inclusion. Differential Equations & Control Processes. 2017;**3**: 15-25

[10] Bouteraa N, Benaicha S. Existence of solutions for nonlocal boundary value

problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling. 2018;**1**(1):45-55

[11] Bouteraa N, Benaicha S. Existence results for fractional differential inclusion with nonlocal boundary conditions. Rivista di Matematica della Università di Parma. 2020;**11**: 181-206

[12] Cernia A. Existence of solutions for a certain boundary value problem associated to a fourth order differential inclusion. International Journal of Analysis and Applications. 2017;**14**:27-33

[13] Ntouyas SK, Etemad S, Tariboon J, Sutsutad W. Boundary value problems for Riemann-Liouville nonlinear fractional diffrential inclusions with nonlocal Hadamard fractional integral conditions. Mediterranean Journal of Mathematics. 2015;**2015**:16

[14] Bouteraa N, Benaicha S. Triple positive solutions of higher-order nonlinear boundary value problems. Journal of Computer Science and Computational Mathematics. June 2017; **7**(2):25-31

[15] Bouteraa N, Benaicha S. Existence of solutions for three-point boundary value problem for nonlinear fractional equations. Analele Universitatii din Oradea. Fascicola Matematica. Tom 2017;**XXIV**(2):109-119

[16] Benaicha S, Bouteraa N. Existence of solutions for three-point boundary value problem for nonlinear fractional differential equations. Bulltin of the Transilvania University of Brasov, Series III: Mathematics, Informtics, Physics. 2017;**10**(2):59

[17] Bouteraa N, Benaicha S. Existence of solutions for third-order three-point boundary value problem. Mathematica. 2018;**60**(1):21-31

[18] Bouteraa N, Benaicha S. The uniqueness of positive solution for higher-order nonlinear fractional differential equation with nonlocal boundary conditions. Advances in the Theory of Nonlinear and Its Application. 2018;**2**(2):74-84

[19] Bouteraa N, Benaicha S, Djourdem H. Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions. Universal Journal of Mathematics and Applications. 2018;**1**(1):39-45

[20] Bouteraa N, Benaicha S. The uniqueness of positive solution for nonlinear fractional differential equation with nonlocal boundary conditions. Analele Universitatii din Oradea. Fascicola Matematica. Tom 2018; **XXV**(2):53-65

[21] Bouteraa N, Benaicha S, Djourdem H, Benatia N. Positive solutions of nonlinear fourth-order twopoint boundary value problem with a parameter. Romanian Journal of Mathematics and Computer Science. 2018;**8**(1):17-30

[22] Bouteraa N, Benaicha S. Positive periodic solutions for a class of fourthorder nonlinear differential equations. Siberian Journal of Numerical Mathematics. 2019;**22**(1):1-14

[23] Bouteraa N. Existence of solutions for some nonlinear boundary value problems [thesis]. Ahmed Benbella, Algeria: University of Oran; 2018

[24] Bouteraa N, Benaicha S, Djourdem H. On the existence and multiplicity of positive radial solutions for nonlinear elliptic equation on bounded annular domains via fixed point index. Maltepe Journal of Mathematics. 2019;**I**(1):30-47

[25] Bouteraa N, Benaicha S, Djourdem H. Positive solutions for systems of fourth-order two-point boundary value problems with parameter. Journal of Mathematical Sciences and Modeling. 2019;**2**(1):30-38

[26] Bouteraa N, Benaicha S. Existence and multiplicity of positive radial solutions to the Dirichlet problem for the nonlinear elliptic equations on annular domains. Studia Universitatis Babeș-Bolyai Mathematica. 2020;**65**(1):109-125

[27] Benaicha S, Bouteraa N, Djourdem H. Triple positive solutions for a class of boundary value problems with integral boundary conditions. Bulletin of Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics. 2020;**13**(1):51-68

[28] Bouteraa N, Djourdem H, Benaicha S. Existence of solution for a system of coupled fractional boundary value problem. Proceedings of International Mathematical Sciences. 2020;**II**(1):48-59

[29] Bouteraa N, Benaicha S. Existence results for second-order nonlinear differential inclusion with nonlocal boundary conditions. Numerical Analysis and Applications. 2021;**14**(1):30-39

[30] Bouteraa N, Inc M, Akinlar MA, Almohsen B. Mild solutions of fractional PDE with noise. Mathematical Methods in the Applied Sciences. 2021:1-15

[31] Bouteraa N, Benaicha S. A study of existence and multiplicity of positive solutions for nonlinear fractional differential equations with nonlocal boundary conditions. Studia

Universitatis Babeș-Bolyai Mathematica. 2021;**66**(2):361-380

[32] Djourdem H, Benaicha S, Bouteraa N. Existence and iteration of monotone positive solution for a fourth-order nonlinear boundary value problem. Fundamental Journal of Mathematics and Applications. 2018;**1**(2):205-211

[33] Djourdem H, Benaicha S, Bouteraa N. Two positive solutions for a fourth-order three-point BVP with signchanging green's function. Communications in Advanced Mathematical Sciences. 2019;**II**(1):60-68

[34] Djourdem H, Bouteraa N. Mild solution for a stochastic partial differential equation with noise. WSEAS Transactions on Systems. 2020;**19**:246-256

[35] Ghorbanian R, Hedayati V, Postolache M, Rezapour SH. On a fractional differential inclusion via a new integral boundary condition. Journal of Inequalities and Applications. 2014; **2014**:20

[36] Inc M, Bouteraa N, Akinlar MA, Chu YM, Weber GW, Almohsen B. New positive solutions of nonlinear elliptic PDEs. Applied Sciences. 2020;**10**:4863. DOI: 10.3390/app10144863

[37] Bouteraa N, Benaicha S. Nonlinear boundary value problems for higherorder ordinary differential equation at resonance. Romanian Journal of Mathematic and Computer Science. 2018;**8**(2):83-91

[38] Bouteraa N, Benaicha S. A class of third-order boundary value problem with integral condition at resonance. Maltepe Journal of Mathematics. 2020; **II**(2):43-54

[39] Lin X, Zhao Z, Guan Y. Iterative technology in a singular fractional

boundary value problem with q-difference. Applied Mathematics. 2016;**7**:91-97

[40] Rezapour SH, Hedayati V. On a Caputo fractional differential inclusion with integral boundary condition for convex-compact and nonconvexcompact valued multifunctions. Kragujevac Journal of Mathematics. 2017;**41**:143-158

[41] Sheng S, Jiang J. Existence and uniqueness of the solutions for fractional damped dynamical systems. Advances in Difference Equations. 2017;**2017**:16

[42] Abbas S, Benchohra M, Bouriah S, Nieto JJ. Periodic solution for nonlinear fractional differential systems. Differential Equations Applications. 2018;**10**(3):299-316

[43] Zhu T. Existence and uniqueness of positive solutions for fractional differential equations. Boundary Value Problems. 2019;**22**:11

[44] Annaby MH, Mansour ZS. q-Fractional calculus and equations. In: Lecture Notes in Mathematics. Vol. 2056. Berlin: Springer-Verlag; 2012

[45] Agrawal O. Some generalized fractional calculus operators and their applications in integral equations. Fractional Calculus and Applied Analysis. 2012;**15**:700-711

[46] Kilbas AA, Srivastava HM, Trijull JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier Science B.V.; 2006

[47] Ahmad B, Ntouyas SK, Alsaedi A. Coupled systems of fractional differential inclusions with coupled boundary conditions. Electronic Journal of Differential Equations. 2019; **2019**(69):1-21

### **Chapter 3**

## Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical Justification, Dynamical Analysis and Neurocomputational Implications

*Serge Gervais Ngueuteu Mbouna*

#### **Abstract**

In this chapter, the dynamical behavior of the incommensurate fractional-order FitzHugh-Nagumo model of neuron is explored in details from local stability analysis. First of all, considering that the FitzHugh-Nagumo model is a mathematical simplification of the Hodgkin-Huxley model, the considered model is derived from the fractional-order Hodgkin-Huxley model obtained taking advantage of the powerfulness of fractional derivatives in modeling certain biophysical phenomena as the dielectrics losses in cell membranes, and the anomalous diffusion of particles in ion channels. Then, it is shown that the fractional-order FitzHugh-Nagumo model can be simulated by a simple electrical circuit where the capacitor and the inductor are replaced by corresponding fractionalorder electrical elements. Then, the local stability of the model is studied using the Theorem on the stability of incommensurate fractional-order systems combined with the Cauchy's argument Principle. At last, the dynamical behavior of the model are investigated, which confirms the results of local stability analysis. It is found that the simple model can exhibit, among others, complex mixed mode oscillations, phasic spiking, first spike latency, and spike timing adaptation. As the dynamical richness of a neuron expands its computational capacity, it is thus obvious that the fractional-order FitzHugh-Nagumo model is more computationally efficient than its integer-order counterpart.

**Keywords:** fractional-order FitzHugh-Nagumo model, fractional derivatives, slow-fast dynamics, mixed mode oscillations, first spike latency

#### **1. Introduction**

When excited sufficiently, the neuron, which is the primary unit of brain and nerves electrical activity, generates an action potential, also known as neuronal

spike, which is a rapid increase then decrease in the neuronal voltage (see for example [1]). This action potential is at the basis of many mechanisms of communication between neurons and therefore is fundamental to understanding signal processing in the brain and nerves activity [2, 3]. Indeed, action potentials can propagate in essentially constant shape away from the cell body along nerves axons and toward synaptic connections with other cells [3]. Neurons exhibit many different spike-based behaviors including regular periodic and chaotic spiking (train of spikes) and bursting (alternation between a quiescent state and repetitive spiking) [3]. Bursting activity via action potentials plays a crucial role in neuronal communication, including robust transmission of signals [4, 5]. Another interesting spikebased behavior is the mixed mode oscillations (MMOs) pattern, which is an alternation between oscillations of distinct large and small amplitudes [6], where large amplitude oscillations are spikes. Certain MMOs patterns are considered as a type of bursting patterns where the period of small amplitude oscillations is considered as the quiescent phase of the bursting pattern.

Since the pioneering work of Hodgkin and Huxley [1], many relevant conductance-based models and simplified phenomenological models have been developed to describe the brain and nerves microscopic dynamical functions. Studying these computational models with the help of the tools of nonlinear dynamics and singular perturbation theory has shed light on the dynamical processes that support the generation of spiking, bursting and MMOs. Indeed, all these models are nonlinear and almost all of them are singularly perturbed systems which are systems involving multiple time scales. The simplest singularly perturbed systems evolve on on two time scales and are therefore named slow-fast systems. Spike as a form of relaxation, bursting and MMOs are the dynamical signatures of the slow-fast property of a system. The complex slow-fast dynamical behaviors that are bursting and MMOs have been intensively investigated in three-dimensional slow-fast systems and have been found previously to occur only in slow-fast systems involving at least three variables because the successive trigger and extinction of spikes suppose the multiple time scale trip of the system trajectory on a complex well organized high-dimensional phase space. A long time afterwards, it was found that noise can induce MMOs in simple two-variable systems as the van der Pol oscillator with constant forcing [7] and FitzHugh-Nagumo-like oscillators [8, 9]. More recently, we found that MMOs can emerge in two-variable systems due to fractional derivation, while studying the fractional-order van der Pol oscillator with constant forcing where the forcing value was set near the Hopf bifurcation point [10]. Subsequently, this result was confirmed by Abdelouahab *et al.* while studying Hopf-like bifurcation and bifurcating oscillatory states in a fractional-order FitzHugh-Nagumo model [11].

Recent studies have shown that the fractional derivation increases the dynamical richness of neuronal models. For example, in Ref. [12], Teka *et al*. have implemented the fractional dynamics on each gating variable of the Hodgkin-Huxley model and they found that the obtained fractional-order versions of the Hodgkin-Huxley model exhibit not only spiking behavior as their integer-order counterpart, but in addition, square wave bursting, MMOs, pseudo-plateau potentials, and phasic spiking. In Ref. [13], Shi and Wang considered a fractional-order Morris-Lecar model and found that this new model exhibits a rich variety of bursting patterns that appear in some common neuron models with properly chosen parameters but do not exist in the corresponding integer-order Morris-Lecar model. In Ref. [14], Mondal *et al*. considered a fractional-order FitzHugh-Rinzel model whose integer-order counterpart is an elliptic burster and found that it exhibits a wide range of neuronal responses including regular spiking, fast spiking, bursting, and MMOs.

#### *Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

In Ref. [15], Teka *et al*. studied the fractional-order Izhikevich model and found that the model produces a wide range of neuronal spike responses, including regular spiking, fast spiking, intrinsic bursting, MMOs, regular bursting and chattering, by adjusting only the fractional derivatives order. The dynamical richness of these fractional-order systems with at least three variables would justify the occurrence of MMOs in the aforementioned two-variable fractional-order slow-fast systems.

In this chapter, we explore further the dynamical behaviors of fractional-order twovariable slow-fast systems by considering an incommensurate fractional-order FitzHugh-Nagumo (FHN) model derived on the basis of biophysical concepts. Commensurate fractional-order FHN models that are widely considered in previous works are just mathematical generalizations that are not supported by any biophysical justification (see for example Ref. [11]). Exploring the behavior of the incommensurate fractional-order FHN model, we have found that depending on the orders of fractional derivatives the model can exhibit two types of MMOs. In the first case, they are identical to classical folded nodes-induced MMOs observed in integer-order systems, also known as canard generated MMOs [16]. In the second case, obtained for lower derivatives orders, the small oscillations of the MMOs pattern start with very low amplitude which then grows slowly before the oscillations enter the spiking phase. This last type of MMOs is identical to singular Hopf bifurcation-induced MMOs observed in integerorder systems [16]. In the second case, the active MMOs phase is sometimes, depending on initial conditions, preceded by a quiescent state which is known in the context of neuroscience as first spike latency. And in addition, the MMOs regime shows a prominent spike timing adaptation as the order of fractional derivatives decrease. The rest of the paper is organized as follows. In Section 2, the incommensurate fractional-order FHN model is derived from the fractional-order Hodgkin-Huxley model obtained using biophysical concepts. In Section 3, it is shown that the fractional-order FHN model can be simulated using an electrical circuit that is similar to the circuit proposed by Nagumo *et al*. [17] with the only difference that the capacitor and the inductor are replaced by corresponding fractional-order electrical elements. Section 4 is devoted to the study of local stability of the fractional-order FHN model, which allows to derive the conditions of occurrence of Hopf-like bifurcation with respect to fractional derivatives orders. In Section 5, the dynamical behavior of the fractional-order FHN model is revealed in order to confirm the results of local stability analysis carried out in the preceding section. A particular attention is granted to MMOs and first spike latency as they are new dynamical features due to fractional derivatives. The chapter ends with a conclusion where the results are summarized.

#### **2. Model and biophysical justification**

The FHN model

$$\begin{aligned} \varepsilon \frac{d\mathbf{x}}{dt} &= \mathbf{y} + \mathbf{x} - \frac{\mathbf{1}}{3} \mathbf{x}^3 + I, \\ \frac{d\mathbf{y}}{dt} &= -\mathbf{x} - \delta \mathbf{y} + \boldsymbol{\gamma}, \end{aligned} \tag{1}$$

is a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley model of the squid giant axon. The derivation of the FHN model as a simple nerve membrane model is based on phase space methods applied to the

Hodgking-Huxley model. Indeed, the phase diagram of the FHN model and a properly chosen projection from the 4-dimensional Hodgking-Huxley phase space onto a plane are similar, which underlines the relationship between the two models.

As the FHN model is just a mathematical representation, to derive its generalization with fractional derivatives, we consider as starting point the generalization of the Hodgking-Huxley model which was derived using biophysical concepts. The Hodgking-Huxley model is given by the following set of differential equations:

$$\begin{aligned} C\_m \frac{dv}{dt} &= I - \overline{g}\_K n^4 (v - v\_\mathcal{K}) - \overline{g}\_{\mathcal{N}\mathcal{A}} m^3 h (v - v\_{\mathcal{N}\mathcal{A}}) - \overline{g}\_l (v - v\_l), \\ \frac{dn}{dt} &= a\_n (1 - n) - b\_n n, \\ \frac{dm}{dt} &= a\_m (1 - m) - b\_m m, \\ \frac{dh}{dt} &= a\_h (1 - h) - b\_h h, \end{aligned} \tag{2}$$

where *v* ¼ *Em* � *Er* is the displacement of the membrane potential *Em* from its resting value *Er*; *n*, *m*, and *h* are the potassium current activation, sodium current activation, and sodium current inactivation gating variables, respectively. *Cm* is the membrane capacity per unit area, *I* is an applied stimulus current. *v*<sup>K</sup> ¼ *E*<sup>K</sup> � *Er*, *v*Na ¼ *E*Na � *Er*, and *vl* ¼ *El* � *Er*, where *E*<sup>K</sup> and *E*Na are the equilibrium potentials for the sodium and potassium ions, and *El* is the potential at which the leakage current due to chloride and other ions is zero. *<sup>g</sup>*K*n*<sup>4</sup> <sup>¼</sup> *<sup>g</sup>*K, *<sup>g</sup>*Na*m*<sup>3</sup>*<sup>h</sup>* <sup>¼</sup> *<sup>g</sup>*Na, where *<sup>g</sup>*K, *<sup>g</sup>*Na and *gl* are ionic conductances. *an*, *bn*, *am*, *bm*, *ah*, *bh* are rate constants which vary with membrane voltage. However, Hodgking and Huxley claimed that "The only major reservation which must be made about [the first equation in Eq. (2): *<sup>I</sup>* <sup>¼</sup> *Cm dv dt* þ *I*Na þ *I*<sup>K</sup> þ *Il*] is that it takes no account of dielectric loss in the membrane" [1]. Now, to account for dielectric losses in a capacitor, Curie proposed the following empiric relation between a DC voltage *U*<sup>0</sup> applied at *t* ¼ 0 and the current *i* it will produce [18]:

$$i(t) = \frac{U\_0}{h\_1 t^a},\tag{3}$$

where *h*<sup>1</sup> is a constant related to the capacitance of the capacitor and the kind of dielectric, 0 <*α* <1, *α* is related to the losses of the capacitor. The lower the losses, the closer to 1 is *α*. Westerlund and Ekstam [19] showed that for a general input voltage *v t*ð Þ, Eq. (3) turns into:

$$\dot{a}(t) = C\_a \frac{d^a v(t)}{dt^a},\tag{4}$$

where *<sup>C</sup><sup>α</sup>* <sup>¼</sup> <sup>Γ</sup>ð Þ <sup>1</sup> � *<sup>α</sup> <sup>=</sup>h*1, *<sup>d</sup>αv t*ð Þ *dt<sup>α</sup>* is the fractional (*α*-order) time derivative of *v t*ð Þ, and Γð Þ� is the Gamma function. On the other hand, Cole claimed that alternating current resistance and capacity measurements over a wide frequency range show that biological materials may be considered electrically equivalent to a circuit including a polarization element considered as a resistance and a capacity in series [20]. When a constant current *i* is started through this element at time *t* ¼ 0, the potential difference across the element may be found to be:

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

$$e(t) = \frac{z\_1 it^a}{\Gamma(1+a)},\tag{5}$$

where *<sup>z</sup>*ð Þ¼ *<sup>ω</sup> <sup>z</sup>*1ð Þ *<sup>j</sup><sup>ω</sup>* �*<sup>α</sup>* is the impedance of the element, with *<sup>j</sup>* <sup>2</sup> ¼ �1. The phase of this element: *φ* ¼ *απ=*2 allowed to provide the value of *α* for diverse biological materials. For example, it was found that *α* is ranged from 0.62 to 0.71 for frog sciatic nerve. This justifies why the power-law dynamics can occur in the membrane electrical activity. Note that Eqs. (3) and (5) are equivalent. Fractional derivatives with power-law kernel have also been used to account for the multiple timescale dynamics in neuroscience i.e. for processes which cannot be dynamically characterized with a single time constant. For example, Lundstrom *et al*. [21] showed that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics and that this multiple time scale adaptation is consistent with fractional-order differentiation, such that the neuron firing rate is a fractional derivative of slowly varying stimulus parameters. Subsequently, Teka *et al*. [22] developed a fractional-order leaky integrate-and-fire model which can reproduce upward and downward spike adaptations found experimentally by Lundstrom *et al*. [21]. In Ref. [22], the fractional derivation operator was applied on the membrane potential. So, the spike timing adaptation would be a dynamical manifestation of dielectric losses in the membrane. So, in order to account for the dielectric losses in the membrane, the total membrane current should be written as follows:

$$I = C\_m^a \frac{d^a v}{dt^a} + I\_{\rm Na} + I\_{\rm K} + I\_{\rm l}, \tag{6}$$

where *C<sup>α</sup> <sup>m</sup>* is a constant related to the capacitance of the membrane. Taking into account the ionic conductances and the gating variables, Eq. (6) is rewritten as follows:

$$\mathbf{C}\_{m}^{a}\frac{d^{a}v}{dt^{a}} = I - \overline{\mathbf{g}}\_{K}n^{4}(v - v\_{\rm K}) - \overline{\mathbf{g}}\_{\rm Na}m^{3}h(v - v\_{\rm Na}) - \overline{\mathbf{g}}\_{l}(v - v\_{l}).\tag{7}$$

On the other hand, it has been shown that the dynamics of the gating variables would better be described using fractional derivatives instead of first-order derivatives [12]. Indeed, the gating dynamics is more complicated than what is traditionally assumed. Let us recall that ion channels are complex membrane proteins which provide ion-conducting, nanoscale pores in the biological membranes [23]. The gating dynamics is the spontaneous conformational dynamics of these proteins resulting in stochastic transitions between the conducting and non-conducting states also known as open and closed states of the pore, respectively. It was considered that the transitions between the open state and the closed state is a Markovian stochastic process that can be characterized by exponential residence time distributions of open and closed time intervals. However, experimental investigations had revealed that the distributions of the residence time intervals of closed states are typically not exponential. For several ion channels, these residence time distributions can be fitted by a stretched exponential function [24], or by a power-law function [25]. In Ref. [23], the time derivative of the Mittag-Leffler function which interpolates between the stretched exponential (at small time) and the asymptotic long time power-law function, was considered to show that the closed state of the ion channel can be modeled as an anomalous diffusion process over a large number of traps, described by a time-fractional diffusion equation. Accordingly, the dynamics of gating variables in the Hodgking-Huxley equation should be described by fractional-order differential equations as shown by Teka *et al*. [12], i.e.:

$$\frac{d^\beta \mathbf{z}}{dt^\beta} = a\_\mathbf{z}(1-\mathbf{z}) - b\_\mathbf{z}\mathbf{z},\tag{8}$$

where **z** ¼ ½ � *n*, *m*, *h* , and *β* is a parameter related to the power-law index of anomalous diffusion. As shown in Ref. [23], the diffusion process over the substates of the closed state is a subdiffusive phenomenon, which corresponds to 0< *β* <1.

Combining Eqs. (7) and (8), one obtains the incommensurate fractional-order Hodgking-Huxley model. Now, the variables *x* and *y* of the FHN model have been considered to describe the membrane voltage and the coarse-grained action of the gating variables, respectively. So, the variables *x* and *y* correspond to *v* in Eq. (7) and **z** in Eq. (8), respectively. Thus, the incommensurate fractional-order Hodgking-Huxley model given by Eqs. (7) and (8) is reduced to the following incommensurate fractional-order FHN model:

$$\begin{aligned} \epsilon D\_t^a \mathbf{x} &= \mathbf{y} + \mathbf{x} - \frac{1}{3} \mathbf{x}^3 + I, \\ D\_t^\beta \mathbf{y} &= -\mathbf{x} - \delta \mathbf{y} + \mathbf{y}, \end{aligned} \tag{9}$$

where *D<sup>α</sup> <sup>t</sup> <sup>x</sup>* <sup>¼</sup> *<sup>d</sup><sup>α</sup> x dt<sup>α</sup>* is the fractional-order (*α*-order) time derivative of *x t*ð Þ defined in the sense of Caputo [26] as follows:

$$D\_t^a \varkappa(t) = \frac{1}{\Gamma(1-a)} \int\_0^t \frac{\varkappa^{(1)}(\tau)}{(t-\tau)^a} d\tau,\tag{10}$$

where *α* ∈ ℝ and 0 <*α* <1, and Γð Þ� is the Gamma function. The Caputo's definition of fractional derivative is suitable for initial value problems. Note that lim*<sup>α</sup>*!1�*D<sup>α</sup> <sup>t</sup> x t*ðÞ¼

*<sup>x</sup>*ð Þ<sup>1</sup> ðÞ¼ *<sup>t</sup> dx t*ð Þ *dt* [27] and then for *α* ! 1 and *β* ! 1, the set of eqs. (9) is reduced to the classical FHN model given by Eq. (1). For *I* ¼ *δ* ¼ 0, Eq. (9) amounts to the incommensurate fractional-order van der Pol oscillator with constant forcing studied by us in Ref. [10]. If we apply the transformation *y* ! �*y* and set *β* ¼ *α* in Eq. (9), we uncover another version of the fractional-order FHN model considered by Abdelouahab *et al*. [11].

#### **3. Equivalent electrical circuit**

An example of electrical circuit capable to reproduce the dynamical behavior of the fractional-order FHN model is apparently the same as the one considered by Nagmo *et al*. [17] as shown in **Figure 1**, where L is a fractional-order inductor, C is a fractional-order capacitor, and the other electronic elements are classic. Since, there is not yet a conventional manner to represent fractional-order electronic elements, we prefer to represent them as classical elements. TD is a tunnel diode whose voltagecurrent characteristic is given by *f e*ð Þ¼ *i*<sup>0</sup> � ð Þ� *e* � *e*<sup>0</sup> ð Þ *e* � *e*<sup>0</sup> 3 *=*3*K*<sup>2</sup> � �*=ρ*, where *i*<sup>0</sup> ¼ *f e*ð Þ<sup>0</sup> . The fractional capacitor and the fractional inductor are characterized by

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

**Figure 1.** *An electrical circuit simulating the fractional-order FHN model.*

*iC* <sup>¼</sup> *<sup>C</sup><sup>α</sup> <sup>d</sup><sup>α</sup> v <sup>d</sup>τα* , and *vL* <sup>¼</sup> *<sup>L</sup><sup>β</sup> <sup>d</sup><sup>β</sup> i <sup>d</sup>τβ* , respectively, where *<sup>C</sup><sup>α</sup>* and *<sup>L</sup><sup>β</sup>* are parameters related to their capacitance and inductance, with 0 <*α* <1, 0<*β* <1. Some of these coefficients can be found in Refs. [19, 28] for real capacitors and in Ref. [29] for real inductors. Applying the Kirchoff's law, it comes out that the circuit of **Figure 1** is described by the following set of fractional-order differential equations:

$$\begin{split} \mathcal{L}^a \frac{d^a v}{d\tau^a} &= i + i\_0 - \frac{1}{\rho} \left( (E\_0 - v - \varepsilon\_0) - \frac{1}{3K^2} (E\_0 - v - \varepsilon\_0)^3 \right) + j, \\ \mathcal{L}^\beta \frac{d^\beta i}{d\tau^\beta} &= -\text{Ri} - v. \end{split} \tag{11}$$

Let *τ<sup>L</sup>* ¼ *L=ρ* and *τ<sup>C</sup>* ¼ *ρC*, the time constants related to the dynamics of the inductor and capacitor, respectively. Let us introduce the following dimensionless variables: *t* ¼ *τ=τL*, *x* ¼ ð Þ *v* þ *e*<sup>0</sup> � *E*<sup>0</sup> *=K*, and *y* ¼ *ρ*ð Þ *i* þ *i*<sup>0</sup> *=K*, and use the fractional differential operator *D<sup>α</sup> <sup>t</sup>* . Then, Eq. (11) can be rewritten as follows:

$$\begin{aligned} \epsilon \pi\_L^{1-a} D\_t^a \varkappa &= \mathcal{y} + \varkappa - \frac{1}{3} \varkappa^3 + I, \\ \pi\_L^{1-\beta} D\_t^\beta \mathcal{y} &= -\varkappa - \delta \mathcal{y} + \mathcal{y}, \end{aligned} \tag{12}$$

where *<sup>ε</sup>* <sup>¼</sup> *<sup>τ</sup>C=τ<sup>L</sup>* <sup>¼</sup> *<sup>ρ</sup>*<sup>2</sup>*C=L*, *<sup>I</sup>* <sup>¼</sup> *<sup>ρ</sup>j=K*, *<sup>δ</sup>* <sup>¼</sup> *<sup>R</sup>=ρ*, *<sup>γ</sup>* <sup>¼</sup> ð Þ *Ri*<sup>0</sup> <sup>þ</sup> *<sup>e</sup>*<sup>0</sup> � *<sup>E</sup>*<sup>0</sup> *<sup>=</sup>K*. The difference of scales between *τ<sup>L</sup>* and *τ<sup>C</sup>* is at the basis of the slow-fast dynamics that results in relaxation oscillations in the FHN system behavior. Indeed, *τ<sup>C</sup>* ≪ *τL*, then *ε* ¼ *τC=τ<sup>L</sup>* ≪ 1 acts as a time scales ratio between *x* and *y*. Without harming any generality, we will consider *τ<sup>L</sup>* ¼ 1, since the only effect of this parameter is to reinforce the relaxation that is expressed yet in *ε*. Then, Eq. (12) can be rewritten without *τL*, which amounts to the fractional-order FHN model given by Eq. (9). So, the fractional-order FHN model can be simulated with the electrical circuit in **Figure 1** where the capacitor and the inductor are fractional-order electrical elements known as fractances.

#### **4. Local stability analysis and Hopf-like bifurcation**

The equilibrium points E *x*<sup>∗</sup> , *y* <sup>∗</sup> � � of Eq. (9) are solutions of the following set of algebraic equations:

$$\begin{aligned} \mathbf{x}\_\*^3 - \mathbf{3} \left( \mathbf{1} - \frac{\mathbf{1}}{\delta} \right) \mathbf{x}\_\* - \mathbf{3} \left( I + \frac{\gamma}{\delta} \right) &= \mathbf{0}, \\ \mathbf{y}\_\* = \frac{-\mathbf{x}\_\* + \gamma}{\delta} .\end{aligned} \tag{13}$$

We will consider the case where this equation admits only one solution, i.e. for �4 1ð Þ � <sup>1</sup>*=<sup>δ</sup>* <sup>3</sup> <sup>þ</sup> <sup>9</sup>ð Þ *<sup>I</sup>* <sup>þ</sup> *<sup>γ</sup>=<sup>δ</sup>* <sup>2</sup> <sup>&</sup>gt;0, according to the Cardan's method. The fractional dynamics does not affect neither the number of equilibrium points nor their positions, but it may change their stability [26]. So, it is proper to study the stability of E in this particular context and conclude about the effect of the fractional derivatives. To do so, we will first consider the local stability of the classical integer-order FHN model. Let f g*λ* be the eigenvalues spectrum of the Jacobian matrix **J**<sup>E</sup> of Eq. (9) evaluated at equilibrium point E *x*<sup>∗</sup> , *y* <sup>∗</sup> � �. The corresponding eigenvalues are conjugate complex numbers given by:

$$
\lambda\_{1,2} = \frac{-\left(\epsilon\delta - 1 + \varkappa\_\*^2\right) \pm j\sqrt{\Delta}}{2\varepsilon},
\tag{14}
$$

where *j* <sup>2</sup> ¼ �1, and <sup>Δ</sup> <sup>¼</sup> <sup>4</sup>*<sup>ε</sup>* <sup>1</sup> � *<sup>δ</sup>* <sup>1</sup> � *<sup>x</sup>*<sup>2</sup> ∗ � � � � � *εδ* � <sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> ∗ � �<sup>2</sup> >0*:* Then, E is a focus, a key ingredient for the occurrence of Hopf bifurcation. According to Eq. (14), E is stable (for the integer-order system) if:

$$
\epsilon \delta - \mathbf{1} + \mathbf{x}\_\*^2 > \mathbf{0}.\tag{15}
$$

Thus, Hopf bifurcations occur at *x*<sup>∗</sup> ¼ *x*<sup>þ</sup> <sup>H</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi *εδ* � <sup>1</sup> <sup>p</sup> and *<sup>x</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>x</sup>*� <sup>H</sup> ¼ � ffiffiffiffiffiffiffiffiffiffiffiffiffi *εδ* � <sup>1</sup> <sup>p</sup> which correspond via Eq. (13) to two values of the stimulus current, namely *I* þ <sup>H</sup> and *I* � H, respectively. **Figure 2** shows the bifurcation diagram computed using Matcont Matlab toolbox, around *I* þ <sup>H</sup> for a set of parameters chosen very close to the one used by FitzHugh in its pioneering work [30], namely *ε* ¼ 0*:*1, *δ* ¼ 0*:*8, *γ* ¼ 0*:*7, that will be used all through the paper. **Figure 2** shows that the Hopf bifurcation obtained for this set of parameters is subcritical.

Secondly, we consider the fractional-order FHN model with different rational orders *α* ¼ *m=m*<sup>0</sup> and *β* ¼ *n=n*<sup>0</sup> , with *m*,*n*,*m*<sup>0</sup> ,*n*<sup>0</sup> ∈ℕ. Let *M* be the less common multiple of *m*<sup>0</sup> and *n*<sup>0</sup> . According to the Theorem on the stability of incommensurate fractionalorder systems [26], the equilibrium point E is asymptotically stable if all the roots *λ<sup>k</sup>* of the following equation:

$$\det\left(\mathbf{diag}\left[\lambda^{Ma}\;\lambda^{M\beta}\right]-\mathbf{J}\_{\mathcal{E}}\right)=\mathbf{0},\tag{16}$$

satisfy the following condition:

$$|\arg(\lambda\_k)| > \frac{\pi}{2M}, \text{ } \forall \ k,\tag{17}$$

where **J**<sup>E</sup> is the Jacobian matrix of the system evaluated at E. The stability condition given by Eq. (17) is equivalent to the following:

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

**Figure 2.**

*One parameter bifurcation diagram of the FHN system with respect to the stimulus current I, for ε* ¼ 0*:*1*, δ* ¼ 0*:*8*, γ* ¼ 0*:*7*. The solid (resp. dashed) gray line depicts stable (resp. unstable) equilibrium point; the black solid (resp. red hollow) circle markers depict the extremums of stable (resp. unstable) periodic orbits. H and LPC label Hopf bifurcation and fold bifurcation of periodic orbits also known as saddle-node bifurcation of limit cycles and limit point bifurcation for cycles, respectively.*

$$\lambda\_k \notin \mathcal{D} = \left\{ z \in \mathbb{C} / z = r \mathbf{e}^{j\theta}, \ 0 \le r < R \to \infty, \ |\theta| \le \frac{\pi}{2M} \right\}, \ \forall \ k. \tag{18}$$

And Eq. (17) is equivalent to the following characteristic equation:

$$P(\lambda) = \lambda^{M(a+\beta)} + \delta\lambda^{Ma} - \frac{\mathbf{1} - \boldsymbol{\varkappa}\_{\ast}^{2}}{\varepsilon}\lambda^{M\beta} + \frac{\mathbf{1} - \delta(\mathbf{1} - \boldsymbol{\varkappa}\_{\ast}^{2})}{\varepsilon} = \mathbf{0}.\tag{19}$$

Let BD be the boundary of D. According to the Cauchy's argument Principle [31, 32], if Eq. (19) has no root on BD the closed curve *P*ð Þ BD encircles the origin *N* times, where *N* is the number of roots of Eq. (19) inside the domain D. Accordingly, the stability condition given by Eq. (18) requires that *N* ¼ 0. Therefore, the stability condition can be resumed in the following theorem [32]:

**Theorem 1**: The equilibrium point E of the fractional-order FHN model is stable if *P*ð Þ BD neither encircles nor gets through the origin O in the complex plan.

Drawing one's inspiration from the method of exploitation of Theorem 1 proposed in Ref. [32] and improved in Ref. [10], one can derive the following stability condition for the equilibrium point E of the incommensurate fractional-order FHN model:

$$\zeta^{a+\beta}\cos\frac{(a+\beta)\pi}{2} + \delta\zeta^a\cos\frac{a\pi}{2} - \frac{1-\varkappa\_\*^2}{\varepsilon}\zeta^\beta\cos\frac{\beta\pi}{2} + \frac{1-\delta\left(1-\varkappa\_\*^2\right)}{\varepsilon} > 0,\tag{20}$$

where *ζ* is solution of the following equation:

$$
\zeta^{a+\beta}\sin\frac{(a+\beta)\pi}{2} + \delta\zeta^a\sin\frac{a\pi}{2} - \frac{1-\varkappa\_\*^2}{\varepsilon}\zeta^\beta\sin\frac{\beta\pi}{2} = 0.\tag{21}
$$

where *<sup>ζ</sup>* <sup>¼</sup> *rM*, with *<sup>r</sup>*<sup>∈</sup> **<sup>ℝ</sup>**þ. For a given value of *<sup>I</sup>*, Eq. (13) is solved and the solution *x*<sup>∗</sup> is introduced into Eq. (21) which is solved its turn for a given set ð Þ *α*, *β* , and the solution *ζ* is introduced into Eq. (20) to check the stability condition. Let us recall that in the case of the integer-order system, the stability changes via Hopf bifurcations. Now, considering its definition, a Hopf bifurcation cannot occur in a fractional-order system which cannot have exact periodic solutions on a finite time interval [33]. However, *S*-asymptotically *T*-periodic functions, can occur as solutions of a fractional-order autonomous system with fixed bounded lower terminal, instead of normal *T*-periodic solutions [34]. Then, the concept of Hopf-like bifurcation has been introduced to characterize the stability change of an equilibrium point giving rise to *S*-asymptotically *T*-periodic solutions [11]. As the stimulus current *I* is the bifurcation parameter, the numerical simulation of the set of Eqs. (13), (20) and (21) shows that the stability of the equilibrium point E of the fractional-order FHN model switches via Hopf-like bifurcations at two critical points that we will refer to as *I* þ <sup>H</sup> and *I* � H, whose values depend on *α* and *β* . It is worthwhile pointing out that for *β* ¼ *α* ! 1 these bifurcation points merge with those obtained above in the case of integer-order FHN model.

In the case, where the fractional dynamics appears only in the membrane potential (i.e., for 0 <*α*< 1 and *β* ! 1), and in the case where the fractional dynamics appears only in the gating variables (i.e., for *α* ! 1 and 0<*β* <1), the stability conditions can be derived easily from Eqs. (20) and (21). These two limiting cases are depicted in **Figures 3** and **4**, where we can see how the positions of the bifurcations points *I* þ <sup>H</sup> and

#### **Figure 3.**

*Stability chart of the equilibrium point* E *in I*ð Þ , *α space for β* ! 1*: (b) is a zoom of (a). The gray colored area is the oscillatory region, which corresponds to unstable* E*. The black solid lines depict the Hopf-like bifurcations or stability boundaries I*<sup>þ</sup> <sup>H</sup>ð Þ *α and I*� <sup>H</sup>ð Þ *α . UE (resp. SE): Unstable (resp. stable) equilibrium point* E*.*

#### **Figure 4.**

*Stability chart of the equilibrium point* E *in I*ð Þ , *β space for α* ! 1*: (b) is a zoom of (a). The gray colored area is the oscillatory region, which corresponds to unstable* E*. The black solid lines depict the Hopf-like bifurcations or stability boundaries I*<sup>þ</sup> <sup>H</sup>ð Þ *β and I*� <sup>H</sup>ð Þ *β . UE (resp. SE): Unstable (resp. stable) equilibrium point* E*.*

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

**Figure 5.** *Effect of α and β on the stability boundary (Hopf-like bifurcation) I*<sup>þ</sup> H*.*

*I* � <sup>H</sup> vary with respect to the fractional derivatives orders *α* and *β*. **Figure 5** shows the effects of the fractional-order derivatives on the stability boundaries of **Figure 4(b)**. **Figure 5** corresponds to the general case where the fractional dynamics appear in both the membrane potential and gating variables, that is, for 0 <*α*< 1 and 0<*β* <1 . Overall, as shown in **Figures 3**–**5**, as the derivatives orders *α* and *β* decrease, the region corresponding to unstable equilibrium shrinks, involving the expansion of stability regions. Thus, as expected, the fractional-order derivation enhances stability in the dynamics of the FHN system.

In what comes next, keeping in mind the above local stability analysis, we will examine the oscillatory behaviors of the fractional-order FHN system.

#### **5. Dynamical behavior and neurocomputational implications**

The incommensurate fractional-order FHN model given by Eq. (9) is solved numerically thanks to the Adams-Bashforth-Moulton predictor-corrector scheme [35], with the set of parameters used for **Figure 2**, that is, *ε* ¼ 0*:*1, *δ* ¼ 0*:*8, *γ* ¼ 0*:*7. Unless otherwise indicated, the initial conditions are set as *x*ð Þ¼ 0 0 and *y*ð Þ¼ 0 0 . The fractional derivatives orders *α* and *β*, and the input stimulus current *I* are assumed to be control parameters.

Let us recall that, for the chosen set of parameters, the dynamics of the classical integer-order FHN model (Eq. (1) or Eq. (9) for *α* ! 1 and *β* ! 1) converges either to the equilibrium point (resting state) or to a limit cycle with relaxation oscillations (spiking state), depending on the strength of the stimulus *I*. Since, the transition between these two regimes occurs via a subcritical Hopf bifurcation, there is an interval of *I* where the two attractors coexist.

In the case, where the fractional dynamics appears only in the membrane potential (i.e. for 0<*α* <1 and *β* ! 1), a narrow region of a new regime, namely phasic spiking, appears between the regions of existence of resting state and spiking state. The phasic spiking pattern is made up of a spiking phase transient to resting. The size of its region of existence in the parameter space increases with decreasing value of *α*. **Figure 6** shows an illustration of these three dynamical regimes. Note that the region of existence of spiking state extends beyond the stability threshold, which means that the subcritical Hopf-like bifurcation persists. Keeping the value of *α* ∈� ½ 0, 1 , the value of *β* is reduced a bit, say from 1 to 0.9. The bifurcation scenario changes a lot. When the system loses stability, MMOs take place (see **Figure 7**) and when the value of *I* decreases further, there is a transition to spiking state. For high values of *α*, at the

**Figure 6.**

*Time series* �*x t*ð Þ *showing the dynamical regimes exhibited by the FHN model with fractional-order dynamics only in the membrane potential (i.e. β* ! 1*) for α* ¼ 0*:*8*, and different values of the stimulus I: (a) I* ¼ �0*:*400*: Spiking; (b) I* ¼ �0*:*345*: Phasic spiking; (c) I* ¼ �0*:*330*: Resting.*

#### **Figure 7.**

*Time series* �*x t*ð Þ *showing some MMOs patterns exhibited by the fractional-order FHN model for α* ¼ 0*:*8*, β* ¼ <sup>0</sup>*:*9*, and different values of I: (a)* <sup>1</sup><sup>15</sup> *pattern for I* ¼ �0*:*4146*; (b)* <sup>1</sup>312 *pattern for I* ¼ �0*:*4290*; (c)* <sup>1</sup><sup>2</sup> *pattern for I* ¼ �0*:*4320*; (d)* <sup>1</sup>121 *pattern for I* ¼ �0*:*4472*.*

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

**Figure 8.**

*Time series* �*x t*ð Þ *showing some dynamical regimes exhibited by the fractional-order FHN model: (a) small amplitude oscillations for α* ¼ 0*:*985*, β* ¼ 0*:*9*, and I* ¼ �0*:*3824*; (b) phasic spiking for α* ¼ 0*:*6*, β* ¼ 0*:*9*, and I* ¼ �0*:*427*.*

transition between resting state and MMOs, a narrow window of small amplitude oscillations (see **Figure 8**) appears. A very narrow region of another type of phasic spiking appears, for relatively small values of *α*, at the transition from resting state to MMOs. This new type of phasic spiking is made up of transient MMOs to resting (see **Figure 8**). The lifetime and the number of spikes of this transient MMOs regime increases as the strength of the stimulus current goes away from the bifurcation point.

Regular MMOs have been often referred to as *L<sup>s</sup>* patterns, where *L* and *s* are the number of large amplitudes and the number small amplitudes in one pattern, respectively. For example, **Figure 7** shows MMOs with 1<sup>15</sup> pattern in (a), and 12 pattern in (c). When the value of the control parameter *I* varies, MMOs with complex patterns develop at the transition between two MMOs states with regular patterns; for example, **Figure 7** shows MMOs with 1<sup>3</sup> 12 pattern in (b); and 1<sup>1</sup> 21 pattern in (d). The 13 12 MMOs develops between 13 MMOs and 1<sup>2</sup> MMOs, and 1<sup>1</sup> 21 MMOs develops between 1<sup>1</sup> MMOs and 21 MMOs. **Figure 7** also shows that when the value of *I* is close to the bifurcation point *I* þ <sup>H</sup><sup>≈</sup> � <sup>0</sup>*:*41185, we have 1*<sup>s</sup>* MMOs – the value of *<sup>s</sup>* decreases for decreasing value of *I*. As the value of *I* decreases further and the value of *s* reaches 1, one obtains complex MMOs patterns with many large amplitude oscillations, which finally leads to *L*<sup>1</sup> MMOs (where *L* increases with decreasing value of *I*). The MMOs patterns shown in **Figure 7** are identical to classical folded nodes-induced MMOs observed in integer-order systems, also known as canard generated MMOs [16].

In order to illustrate the bifurcation scenarios described above, we map all the dynamical regimes in the parameter space as shown in **Figure 9**. As mentioned above, the Hopf-like bifurcation obtained there is subcritical as shown in **Figure 9(a)**. Indeed, there is a parameter region where oscillatory states coexist with resting state. However, **Figure 9(b)** show that when the value of *β* decreases, the limit between oscillatory states and resting state is marked clearly by the stability threshold obtained by local stability analysis.

We now examine the case where the fractional dynamics appears only in the gating variables, that is, for *α* ! 1 and 0 <*β* <1 . The resting state exists for high values of *I*. As the value *I* decreases the equilibrium state loses stability and MMOs take place. At the transition between resting state and MMOs, a very narrow window of small amplitude oscillations appears. As the value of *I* decreases further, there is a transition from MMOs to spiking state. Keeping 0 <*β* <1, the value of *α* is reduced a bit, say from 1 to 0.9. The bifurcation scenario does not change significantly. Finally, we map

#### **Figure 9.**

*Dynamical regimes maps in I*ð Þ , *α space superimposed on the stability boundaries I*<sup>þ</sup> <sup>H</sup>ð Þ *α obtained analytically: (a) β* ! 1*; (b) β* ¼ 0*:*9*. Gray colored region: Spiking; cyan colored region: MMOs; magenta colored region: Small amplitude oscillations (SAOs); red colored region: Phasic spiking (PS), white region: Resting. The square markers in (a), circle markers and diamond marker in (b) show the parameters used for Figures 6–8, respectively.*

all the dynamical behaviors in the parameter space as shown in **Figure 10**. One can notice that the result of the global dynamics analysis obtained numerically agrees very well with the local stability analysis result. This figure shows that the domain of existence of MMOs widens as the value of *β* decreases. The major difference between the cases depicted in the subsets (a) and (b) of **Figure 10** is that the region of existence of small amplitude oscillations shrinks with decreasing value of *α*.

These phase diagrams in **Figures 9** and **10** show the rich variety of dynamical behaviors of the fractional-order FHN neuron model.

For low values of *β* (say, lesser than about 0.7) one observes another type of MMOs for which small oscillations start with very low amplitude which then grows slowly before the oscillations enter the spiking phase (see **Figure 11**). This last type of MMOs is identical

#### **Figure 10.**

*Dynamical regimes maps in I*ð Þ , *β space superimposed on the stability boundaries I*<sup>þ</sup> <sup>H</sup>ð Þ *β obtained analytically: (a) α* ! 1*; (b) α* ¼ 0*:*9*. Gray colored region: Spiking; cyan colored region: MMOs; magenta colored region: Small amplitude oscillations (SAOs); white region: Resting. The circle marker in (b) show the parameters used for Figure 11.*

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

#### **Figure 11.**

*Time series* �*x t*ð Þ *showing two different behaviors of the fractional-order FHN model for α* ¼ 0*:*9*, β* ¼ 0*:*6*, and I* ¼ �0*:*540 *and different initial conditions: (a) x*ð Þ¼ ð Þ 0 , *y*ð Þ 0 ð Þ 0, 0 *; (b) x*ð Þ¼ ð Þ 0 , *y*ð Þ 0 ð Þ 0, 1 *.*

to singular Hopf bifurcation-induced MMOs observed in integer-order systems [16]. Note that unlike the integer-order FHN model for which in the unstable equilibrium region, the dynamics of the system converges to a unique limit cycle no matter the initial conditions, the dynamics of its fractional-order counterparts is sensitive to initial conditions [11]. The same phenomenon was observed in the fractional-order van der Pol oscillator with constant forcing [10] which is closely related to the FHN model. So, changing the initial conditions, it is possible to uncover new dynamical states. For example, we have change the set of initial conditions ð Þ *x*ð Þ 0 , *y*ð Þ 0 in **Figure 11** from 0, 0 ð Þ to 0, 1 ð Þ, and we have found that the active state is preceded by a static-like regime. This static-like transient state was found for the first time in the context of dynamical systems by us, while studying a fractional-order van der Pol oscillator with constant forcing [10]. In the context of neuroscience, this quiescent transient regime is known as first spike latency. Teka *et al*. [22] found that the fractional derivation also induces the occurrence of first spike latency in a leaky integrate-and-fire model, and that this first spike latency is reinforced by decreasing value of the fractional derivative. On the other hand, still regarding the type MMOs shown in **Figure 11**, the instantaneous spike frequency (the inverse of the duration of the corresponding interspike interval), also known as the firing rate, increases as a function of the interspike interval number and approaches an asymptotic value after a certain number of intervals. Also, the increasing rate of the firing rate decreases with decreasing value of the fractional derivatives orders. This property of neurons known as upward spike timing adaptation, observed experimentally [21], has proved to appear only in fractional-order models [21, 22].

It was found in Ref. [10] that the lifetime of this static-like transient and the pseudo-period (corresponding to the interspike interval) of the asymptotic MMOs increase exponentially with the closeness to the Hopf bifurcation point. The same result is found in the present work. **Figure 12** shows the behavior of the first spike latency (FSL) and of the fiftieth interspike interval (ISI) – the period of the asymptotic MMOs after spike timing adaptation - when the value of the stimulus current *I* varies, coming close to the bifurcation point *I* þ H. The results from the direct numerical simulation of Eq. (9) (depicted in **Figure 12**) have been fitted using the following exponential functions in EzyFit Matlab toolbox:

**Figure 12.**

*First spike latency (FSL) and interspike interval (ISI) versus stimulus current I, for α* ¼ 0*:*9 *and β* ¼ 0*:*6 *. With I* þ *<sup>H</sup>*≈0*:*504497*.*

#### **Figure 13.**

*First spike latency (FSL) versus displacement of the stimulus current I from the Hopf-like bifurcation point I*<sup>þ</sup> <sup>H</sup>*, for α* ¼ 0*:*9 *and different values of β. Note that I*<sup>þ</sup> <sup>H</sup>ð Þ *β* ¼ 0*:*6 ≈ � 0*:*504497 *and I*<sup>þ</sup> <sup>H</sup>ð Þ *β* ¼ 0*:*8 ≈ � 0*:*431551*.*

$$FSL(I) = a \exp\left(\frac{b}{I - I\_{\rm H}^{+}}\right) + c, \quad ISI(I) = a' \exp\left(\frac{b'}{I - I\_{\rm H}^{+}}\right) + c'. \tag{22}$$

For the two cases of fitting, the value of the correlation coefficient is over 0.999, indicating good fittings. Thus, this confirms the exponential growth of the first spike latency and of the interspike interval. Furthermore, as shown in **Figure 13**, the features of the first spike latency and of the interspike interval are reinforced by decreasing values of the orders of fractional derivatives.

Although the specific role of MMOs among the plethora of slow-fast dynamical behaviors occurring in neural systems has not yet been determined, it has been nevertheless suggested that the dynamical richness of a neuron expands its computational capacity by increasing its coding capacity [12, 15, 36]. On the other hand, it has been suggested that first spike latency could code for stimulus recognition in several sensory systems [37–40]. Indeed, the variation of first spike latency with stimulus parameters contains considerable information about those parameters [40]. The first spike latency has also been suggested as a source of information for accurate decisions [22]. In addition, the behavior of the firing rate (or interspike interval) of neurons also provides information on the stimulus statistics [21]. The fractional-order FHN model is a mathematically simple model with complex dynamics features, thus increasing the amount of information that

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

can describe the input. So, the fractional FHN model is more computationally efficient than its integer-order counterpart.

#### **6. Conclusion**

In this work, the dynamical behavior of an incommensurate fractional-order FitzHugh-Nagumo model of neuron has been investigated in details. First of all, the considered fractional-order model has been derived from the fractional-order Hodgkin-Huxley model obtained taking advantage of the powerfulness of fractional derivatives in modeling the dielectric losses in cell membranes, and the anomalous diffusion of particles in ion channels. Then, it has been shown that the fractional-order FitzHugh-Nagumo model can be simulated by a simple electrical circuit. Then, the local stability of the incommensurate fractional-order model has been studied with a particular attention granted to the effect of the fractional derivatives. It has been found that the fractional derivatives enhance the local stability of the model. At last, the dynamical behavior of the fractional-order models has been explored numerically, which has confirmed the results of local stability. It has been found that the fractional-order FitzHugh-Nagumo exhibits a lot of complex dynamical features that are not observed in the behavior of its integer-order counterpart, and that cannot be observed in integer-order two-variable systems. Among others, the fractional-order model exhibits mixed mode oscillations, phasic spiking, first spike latency, and spike timing adaptation. These complex features of the dynamical behavior of the fractional-order model increase the computational capacity of the FitzHugh-Nagumo model. An outlook of this work would be the study of the mechanism(s) underlying the formation of mixed mode oscillations and first spike latency as effects of fractional derivation in two-variable systems.

#### **Conflict of interest**

The author declares no conflict of interest.

#### **Author details**

Serge Gervais Ngueuteu Mbouna Laboratory of Modeling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaoundé I, Yaoundé, Cameroon

\*Address all correspondence to: ngueut@yahoo.fr

© 2022 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**

[1] Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology. 1952;**117**:500-544. DOI: 10.1113%2Fjphysiol.1952.sp004764

[2] Rieke F, Warland D, de Ruyter van Steveninck R, Bialek W. Spikes: Exploring the Neural Code. Cambridge, MA: MIT Press; 1999

[3] Izhikevich EM. Neural excitability, spiking, and bursting. International Journal of Bifurcation and Chaos. 2000; **10**:1171-1266. DOI: 10.1142/S0218 127400000840

[4] Izhikevich EM, Desai NS, Walcott EC, Hoppensteadt FC. Bursts as a unit of neural information: Selective communication via resonance. Trends in Neurosciences. 2003;**26**:161-167. DOI: 10.1016/s0166-2236(03)00034-1

[5] Lisman J. Bursts as a unit of neural information: Making unreliable synapses reliable. Trends in Neurosciences. 1997; **20**:38-43. DOI: 10.1016/s0166-2236(96) 10070-9

[6] Desroches M, Guckenheimer J, Krauskopf B, Kuehn C, Osinga HM, Wechselberger M. Mixed-mode oscillations with multiple time scales. SIAM Review. 2012;**54**:211-288. DOI: 10.1137/100791233

[7] Muratov V-EE. Noise-induced mixedmode oscillations in a relaxation oscillator near the onset of a limit cycle. Chaos. 2008;**18**:015111. DOI: 10.1063/1.2779852

[8] Borowski P, Kuske R, Li Y-X, Cabrera JL. Characterizing mixed mode oscillations shaped by noise and bifurcation structure. Chaos. 2010;**20**: 043117. DOI: 10.1063/1.3489100

[9] Makarov VA, Nekorkin VI, Velarde MG. Spiking behavior in a noisedriven system combining oscillatory and excitatory properties. Physical Review Letters. 2001;**86**:3431-3434. DOI: 10.1103/PhysRevLett.86.3431

[10] Ngueuteu GSM, Yamapi R, Woafo P. Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity. Nonlinear Dynamics. 2014;**78**:2717-2729. DOI: 10.1007/ s11071-014-1620-x

[11] Abdelouahab M-S, Lozi R, Chen G. Complex canard explosion in a fractional-order FitzHugh-Nagumo model. International Journal of Bifurcation and Chaos. 2019;**29**:1950111. DOI: 10.1142/S0218127419501116

[12] Teka W, Stockton D, Santamaria F. Power-law dynamics of membrane conductances increase spiking diversity in a Hodgkin-Huxley model. PLoS Computational Biology. 2016;**12**:e1004776. DOI: 10.1371/journal.pcbi.1004776

[13] Shi M, Wang Z. Abundant bursting patterns of a fractional-order Morris-Lecar neuron model. Communications in Nonlinear Science and Numerical Simulation. 2014;**19**:1956-1969. DOI: 10.1016/j.cnsns.2013.10.032

[14] Mondal A, Sharma SK, Upadhyay RK, Mondal A. Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Scientific Reports. 2019;**9**:15721. DOI: 10.1038/s41598-019- 52061-4

[15] Teka WW, Upadhyay RK, Mondal A. Spiking and bursting patterns of fractional-order Izhikevich model.

*Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical… DOI: http://dx.doi.org/10.5772/intechopen.107270*

Communications in Nonlinear Science and Numerical Simulation. 2018;**56**: 161-176. DOI: 10.1016/j. cnsns.2017.07.026

[16] Curtu R. Singular Hopf bifurcations and mixed-mode oscillations in a twocell inhibitory neural network. Physica D: Nonlinear Phenomena. 2010;**239**: 504-514. DOI: 10.1016/j.physd.2009. 12.010

[17] Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon. Proceedings of the IRE. 1962;**50**:2061-2070. DOI: 10.1109/JRPROC.1962.288235

[18] Westerlund S. Dead matter has memory! Physica Scripta. 1991;**43**: 174-179. http://iopscience.iop.org/ 1402-4896/43/2/011

[19] Westerlund S, Ekstam L. Capacitor theory. IEEE Transactions on Dielectrics and Electrical Insulation. 1994;**1**:826-839. DOI: 10.1109/94.326654

[20] Cole KS. Alternating current conductance and direct current excitation of nerve. Science. 1934;**79**: 164-165. DOI: 10.1126/science.79. 2042.164

[21] Lundstrom BN, Higgs MH, Spain WJ, Fairhall AL. Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience. 2008;**11**: 1335-1342. DOI: 10.1038%2Fnn.2212

[22] Teka W, Marinov TM, Santamaria F. Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Computational Biology. 2014;**10**: e1003526. DOI: 10.1371/journal. pcbi.1003526

[23] Goychuk I, Hänggi P. Fractional diffusion modeling of ion channel gating. Physical Review E. 2004;**70**:051915. DOI: 10.1103/PhysRevE.70.051915

[24] Liebovitch LS, Sullivan JM. Biophysical Journal. Fractal analysis of a voltage-dependent potassium channel from cultured mouse hippocampal neurons. 1987;**52**:979-988. DOI: 10.1016/ S0006-3495(87)83290-3

[25] Millhauser GL, Salpeter EE, Oswald RE. Diffusion models of ion-channel gating and the origin of power-law distributions from single-channel recording. Proceedings of the National Academy of Sciences of the United States of America. 1988;**85**:1503-1507. DOI: 10.1073/pnas.85.5.1503

[26] Caponetto R, Dongola R, Fortuna L, Petráš I. Fractional Order Systems: Modeling and Control Applications. Singapore: World Scientific Publishing Co. Pte. Ltd.; 2010

[27] Li C, Deng W. Remarks on fractional derivatives. Applied Mathematics and Computation. 2007;**187**:777-784. DOI: 10.1016/j.amc.2006.08.163

[28] Faraji S, Tavazoei MS. The effect of fractionality nature in differences between computer simulation and experimental results of a chaotic circuit. Central European Journal of Physics. 2013;**11**:836-844. DOI: 10.2478/s11534- 013-0255-8

[29] Schäfer I, Krüger K. Modelling of lossy coils using fractional derivatives. Journal of Physics D: Applied Physics. 2008;**41**:045001. DOI: 10.1088/0022- 3727/41/4/045001

[30] FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal. 1961;**1**:445-466. DOI: 10.1016/S0006- 3495(61)86902-6

[31] Ahlfors LV. Complex Analysis. 2nd ed. New York: McGraw-Hill; 1966

[32] Tavazoei MS, Haeri M, Attari M, Bolouki S, Siami M. More details on analysis of fractional-order Van der pol oscillator. Journal of Vibration and Control. 2009;**15**:803-819. DOI: 10.1177%2F1077546308096101

[33] Tavazoei MS, Haeri M. A proof for non existence of periodic solutions in time invariant fractional order systems. Automatica. 2009;**45**:1886-1890. DOI: 10.1016/j.automatica.2009.04.001

[34] Henriquez HR, Pierri M, Taboas P. On *S*-asymptotically ω-periodic functions on Banach spaces and applications. Journal of Mathematical Analysis and Applications. 2008;**343**: 1119-1130. DOI: 10.1016/j.jmaa.2008. 02.023

[35] Diethelm K, Ford NJ, Freed D. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics. 2002;**29**:3-22. DOI: 10.1023/ A:1016592219341

[36] Ghosh S, Mondal A, Ji P, Mishra A, Dana SK, Antonopoulos CG, et al. Emergence of mixed mode oscillations in random networks of diverse excitable neurons: The role of neighbors and electrical coupling. Frontiers in Computational Neuroscience. 2020;**14**: 49. DOI: 10.3389/fncom.2020.00049

[37] Gollisch T, Meister M. Rapid neural coding in the retina with relative spike latencies. Science. 2008;**319**:1108-1111. DOI: 10.1126/science.1149639

[38] Johansson RS, Birznieks I. First spikes in ensembles of human tactile afferents code complex spatial fingertip events. Nature Neuroscience. 2004;**7**: 170-177. DOI: 10.1038/nn1177

[39] Chase SM, Young ED. First-spike latency information in single neurons increases when referenced to population onset. Proceedings of the National Academy of Sciences of the United States of America. 2007;**104**:5175-5180. DOI: 10.1073/pnas.0610368104

[40] Heil P. First-spike latency of auditory neurons revisited. Current Opinion in Neurobiology. 2004;**14**: 461-467. DOI: https://psycnet.apa.org/ doi/10.1016/j.conb.2004.07.002

#### **Chapter 4**

## Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces

*Bilal Bilalov, Sabina Sadigova and Zaur Kasumov*

#### **Abstract**

In this chapter an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a secondorder elliptic equation, and we study the Fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is also considered one spectral problem for a discontinuous second-order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

**Keywords:** non-standard function spaces, grand-Sobolev spaces, space of traces, Schauder type estimates, Riesz potentials, elliptic equations, Fredholmness, spectral problem, basicity, Morrey space

#### **1. Introduction**

Differential (also elliptic) equations play an especial role in the study of various processes and phenomena in natural science. Solvability problems of elliptic equations have a very rich history and remarkable monographs by various famous mathematicians are devoted to them. The theory of elliptic equations was developed in an comprehensive way in Hölder classes (solution in the classical sense) and in Hilbertian Sobolev spaces *W<sup>k</sup>* <sup>2</sup>ð Þ Ω . In the above mentioned case, depending on the nature of the problem, there are various methods of solution (for instance, the method of potentials, the periodic case, the method of the theory of functions, spectral method, etc.), which cannot be said for the non-Hilbert case *W<sup>k</sup> <sup>p</sup>*ð Þ Ω , *p* 6¼ 2, in which each method faces certain difficulties. All considered spaces are separable Banach spaces and infinitely differentiable and finite functions are dense in them.

In the study of solvability of differential equations these facts are significant. Note that one of the methods to solve differential equations is a spectral method. To justify the solution by this method, one should study the basis properties of the root vectors of the considered spectral problem in the appropriate Banach function space.

In connection with applications in problems of mechanics, mathematical physics and pure mathematics, the so-called non-standard spaces of functions have greatly increased and the list of such spaces includes Lebesgue spaces with a variable summability index, Morrey spaces, grand Lebesgue spaces, Orlicz spaces, etc. For more details one can see the monographs [1–6]. Compared with other areas of mathematics, the apparatus of harmonic analysis has been fairly well studied in relation to these spaces. The problems of analysis and approximation theory have been relatively well studied in Lebesgue spaces with variable summability index and Morrey spaces (see [7–14]). The above mentioned problems have begun to be studied in Grand Lebesgue spaces, and valuable results have been obtained in this direction (see [15, 16]). The solvability problems of partial differential equations have also begun to be studied in the Sobolev spaces generated by these spaces (see [17–27]). Morrey-Sobolev and grand-Sobolev spaces are not separable and therefore infinitely differentiable and finite functions are not dense in them, in this reason the study the problems of solvability of differential equations in these spaces are of special scientific interest. Therefore, it is necessary to extract reasonable subspaces dictated by differential equations and develop an instruments for studying the solvability of differential equations in these subspaces.

An *m*-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for *m*-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation and we study the fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is considered one spectral problem for a discontinuous second order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

#### **2. Needful information**

#### **2.1 Standard notation**

*Z*<sup>þ</sup> will be the set of non-negative integers. 1, *n* ¼ 1, 2, … , *n*. is the set of complex numbers. *Br*ð Þ¼ *<sup>x</sup>*<sup>0</sup> *<sup>x</sup>*∈*R<sup>n</sup>* f g : j j *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> <sup>&</sup>lt;*<sup>r</sup>* will denote the open ball in *<sup>R</sup><sup>n</sup>* centered at *<sup>x</sup>*0, where j j *<sup>x</sup>* <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *x*2 <sup>1</sup> þ … þ *x*<sup>2</sup> *n* <sup>p</sup> , *<sup>x</sup>* <sup>¼</sup> ð Þ *<sup>x</sup>*1, … , *xn* . h i �; � is a scalar product in *Rn*. *mes M*ð Þ will stand for the Lebesgue measure of the set *M*; ∂Ω will be the boundary of the domain Ω; Ω ¼ Ω⋃ ∂Ω; *diam* Ω will stand for the diameter of the set Ω; *f =<sup>M</sup>* denotes the restriction of *f* to *M*. *C*<sup>∞</sup> <sup>0</sup> ð Þ� Ω will denote the space of infinitely differentiable and finite functions in <sup>Ω</sup> and *<sup>C</sup>*ð Þ *<sup>m</sup>* ð Þ� <sup>Ω</sup> will stand for the space of *<sup>m</sup>*-th order continuously *Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

differentiable functions in the domain Ω. *C<sup>m</sup>* <sup>0</sup> ð Þ� Ω will stand for the space of *m*-th order continuously differentiable and finite functions in the domain Ω. *DL*�will stand for the domain of the operator *L*; *RT*�will stand for the range of the operator *T*; *KerT*� is the kernel of the operator *<sup>T</sup>*; *<sup>T</sup>*<sup>∗</sup> is the adjoint of *<sup>T</sup>*; ½ � *<sup>X</sup>*; *<sup>Y</sup>* is a Banach space of bounded operators acting from *X* to *Y*; ½ �¼ *X* ½ � *X*; *X* . Throughout this paper, *q*<sup>0</sup> will denote the conjugate of a number, i.e. <sup>1</sup> *<sup>q</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>*<sup>0</sup> ¼ 1.

#### **2.2 Elliptic operator of** *m***-th order and some necessary facts**

Let Ω ⊂ *Rn* be some bounded domain with the rectifiable boundary ∂Ω. We will use the notations of [19]. *α* ¼ ð Þ *α*1, … , *α<sup>n</sup>* will be the multiindex with the coordinates *<sup>α</sup><sup>k</sup>* <sup>∈</sup>*Z*þ, <sup>∀</sup>*<sup>k</sup>* <sup>¼</sup> 1, *<sup>n</sup>*; *<sup>∂</sup><sup>i</sup>* <sup>¼</sup> *<sup>∂</sup> xi* will denote the differentiation operator, *<sup>∂</sup><sup>α</sup>* <sup>¼</sup> *<sup>∂</sup>α*<sup>1</sup> <sup>1</sup> *<sup>∂</sup>α*<sup>2</sup> <sup>2</sup> … *<sup>∂</sup>α<sup>n</sup> n* . For every *<sup>ξ</sup>* <sup>¼</sup> *<sup>ξ</sup>*1, … , *<sup>ξ</sup><sup>n</sup>* ð Þ we assume *ξα* <sup>¼</sup> *<sup>ξ</sup>α*<sup>1</sup> 1 *ξ α*2 <sup>2</sup> … *ξα<sup>n</sup> <sup>n</sup>* . Let *L* be an elliptic differential operator of *m*-th order

$$L = \sum\_{|p| \le m} a\_p(\mathbf{x}) \partial^p,\tag{1}$$

where *p* ¼ *p*1, … , *pn* � �, *pk* <sup>∈</sup>*Z*þ, <sup>∀</sup>*<sup>k</sup>* <sup>¼</sup> 1, *<sup>n</sup>*, *ap*ð Þ� <sup>∈</sup>*L*∞ð Þ <sup>Ω</sup> are real functions. Consider the elliptic operator *L*0:

$$L\_0 = \sum\_{|p|=m} a\_p^0 \,\partial^p,\tag{2}$$

with the constant coefficients *a*<sup>0</sup> *<sup>p</sup>* and denote by *J*ð Þ� a fundamental solution of Eq. (2) [28].

Let *L* be an elliptic operator and consider a "tangential operator"

$$L\_{\mathbf{x}\_0} = \sum\_{|p|=m} a\_p \left( \mathbf{x}\_0 \right) \partial^p,\tag{3}$$

at every point *x*<sup>0</sup> ∈ Ω. Denote by *Jx*<sup>0</sup> ð Þ� the fundamental solution of the equation *Lx*0*φ* ¼ 0. The function *Jx*<sup>0</sup> ð Þ� is called a parametrics for the equation *Lφ* ¼ 0 with a singularity at the point *x*0. Let

$$S\_{\mathbf{x}\_0}\boldsymbol{\rho} = \boldsymbol{\upmu}(\mathbf{x}) = \int J\_{\mathbf{x}\_0}(\mathbf{x} - \mathbf{y}) \, \boldsymbol{\uprho}(\mathbf{y}) d\mathbf{y},\tag{4}$$

and

$$T\_{\mathbf{x}\_0} = \mathbb{S}\_{\mathbf{x}\_0}(L\_{\mathbf{x}\_0} - L). \tag{5}$$

Denote the operators *Sx*<sup>0</sup> , *Lx*<sup>0</sup> and *Tx*<sup>0</sup> , corresponding to the point *x*<sup>0</sup> ¼ 0, by *S*0, *L*<sup>0</sup> and *T*0, respectively.

Let us give the definition of smooth boundary.

**Definition 1.1** We will say that the boundary *∂Ω* of a domain *Ω* ⊂ *Rn* belongs to class *C*ð Þ*<sup>k</sup>* if each sufficiently small piece of it can be mapped onto a segment of the hyperplane *xn* <sup>¼</sup> 0 using a coordinate transformation *y x*ð Þ¼ *<sup>y</sup>*1ð Þ *<sup>x</sup>* ; … ; *yn*ð Þ *<sup>x</sup>* � � with a positive Jacobian so that *yi* <sup>∈</sup>*C*ð Þ*<sup>k</sup>* , <sup>∀</sup>*<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>*.

#### **2.3 Grand-Sobolev spaces** *W<sup>m</sup> <sup>q</sup>*Þð Þ **<sup>Ω</sup> and** *WN<sup>m</sup> <sup>q</sup>*Þð Þ **Ω**

Define the grand-Lebesgue space *Lq*Þð Þ Ω , 1<*q*< þ ∞ (throughout this paper we will assume that this condition holds on *q*). Grand-Lebesgue space *Lq*Þð Þ Ω is a Banach space of (Lebesgue) measurable functions *f* on Ω with the norm

$$||f||\_{q} = \sup\_{0 < \varepsilon < q - 1} \left( \varepsilon \int\_{\mathfrak{A}} |f|^{q-\varepsilon} d\mathfrak{x} \right)^{\frac{1}{q-\varepsilon}}.\tag{6}$$

The following continuous embeddings hold

$$L\_q(\mathfrak{Q}) \subset L\_q(\mathfrak{Q}) \subset L\_{q-\varepsilon}(\mathfrak{Q}),\tag{7}$$

where *ε*∈ ð Þ 0, *q* � 1 is an arbitrary number. The space *Lq*Þð Þ Ω is not separable.

Below in this section we will assume that every function defined on Ω is extended by zero to *<sup>R</sup><sup>n</sup>*nΩ. Let *<sup>T</sup><sup>δ</sup>* be a shift operator, i.e. ð Þ *<sup>T</sup>δ<sup>f</sup>* ð Þ¼ *<sup>x</sup> <sup>f</sup>*ð Þ *<sup>δ</sup>* <sup>þ</sup> *<sup>x</sup>* , <sup>∀</sup>*x*<sup>∈</sup> <sup>Ω</sup> and ∀*δ*∈*R<sup>n</sup>*. Let

$$N\_{q\rangle}(\mathfrak{Q}) = \left\{ f \in L\_q(\mathfrak{Q}) : \left\| T\_{\delta}f - f \right\|\_{q\rangle} \to \mathfrak{0}, \delta \to \mathfrak{0} \right\}.\tag{8}$$

The space *Nq*Þð Þ Ω is a Banach space with the norm k k� *<sup>q</sup>*Þ, (i.e. is the subspace of *Lq*Þð Þ Ω .)

The following lemma is true (see [17]).

**Lemma 1.2** *C*<sup>∞</sup> <sup>0</sup> ð Þ¼ *Ω Nq*Þð Þ *Ω* (the closure is taken in *Lq*Þð Þ *Ω* ). Let us include the following lemma without proof. **Lemma 1.3** *The embeddings*

$$L\_q(\mathfrak{Q}) \subset N\_q(\mathfrak{Q}) \subset L\_q(\mathfrak{Q}) \subset L\_1(\mathfrak{Q}),\tag{9}$$

hold and every inclusion is strict.

Denote by *W<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> the grand-Sobolev space generated by the norm

$$||f||\_{W\_{q)}^m} = \sum\_{k=0}^m \left\| \left. f^{(k)} \right\|\_{q} \right\|\_{q}. \tag{10}$$

Let

$$\mathcal{W}\mathcal{W}\_{q\rangle}^{m}(\boldsymbol{\Omega}) = \left\{ f \in \mathcal{W}\_{q\rangle}^{m}(\boldsymbol{\Omega}) : \||T\_{\boldsymbol{\delta}}f - f||\_{\mathcal{W}\_{q\rangle}^{m}} \to \mathbf{0}, \boldsymbol{\delta} \to \mathbf{0} \right\}.\tag{11}$$

Consider the following singular kernel

$$k(\mathbf{x}) = \frac{o(\mathbf{x})}{|\mathbf{x}|^n},\tag{12}$$

where *ω*ð Þ *x* is a positive homogeneous function of degree zero, which is infinitely differentiable and satisfies

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

$$\int\_{|\mathfrak{x}|=1} a(\mathfrak{x}) d\sigma = \mathbf{0},\tag{13}$$

*dσ* being a surface element on the unit sphere. By *K* we will denote the corresponding singular integral

$$(Kf)(\infty) = k \ast f(\infty) = \int\_{\Omega} f(\mathfrak{y}) k(\mathfrak{x} - \mathfrak{y}) d\mathfrak{y}.\tag{14}$$

The following theorem is valid for the operator *K* (see [4]).

**Theorem 1.4** [4] *The inclusion K* <sup>∈</sup> *Lq*Þð Þ *<sup>Ω</sup>* � �, 1<*q*<sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> *is valid, i.e.* <sup>∃</sup>*c*>0*:* k k *Kf <sup>q</sup>*<sup>Þ</sup> <sup>≤</sup> *c f* k k*<sup>q</sup>*Þ, <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Lq*Þð Þ <sup>Ω</sup> (15)

The validity of the following lemma is given in [17].

**Lemma 1.5** [17] *Nq*Þð Þ *Ω* , 1< *q*< þ ∞, is an invariant subspace of the singular operator *K* in *Lq*Þð Þ *Ω* .

Considering the expression for the norm *Nq*Þ, it is not hard to prove the following.

**Proposition 1.6** *Let <sup>Ω</sup>* <sup>⊂</sup>*R<sup>n</sup>*�*be a bounded domain and L be an elliptic operator with coefficients ap* <sup>∈</sup>*L*∞ð Þ *<sup>Ω</sup>* , <sup>∀</sup>j j *<sup>p</sup>* <sup>≤</sup> *m. Then it is valid L*<sup>∈</sup> *WN<sup>m</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* ; *Nq*Þð Þ *<sup>Ω</sup>* h i*, i.e. the following inequality*

$$||Lu||\_{N\_{q}(\Omega)} \leq C||u||\_{\mathcal{W}N\_{q}^{m}(\Omega)}, \forall u \in \mathcal{W}N\_{q}^{m}(\Omega),\tag{16}$$

holds, where *C* >0 is a constant independent of *u*.

In the sequel, when Ω ¼ *Br* the spaces *Lq*Þð Þ Ω , *Nq*Þð Þ Ω ,*Wq*Þð Þ Ω and *WNq*Þð Þ Ω will be redenoted by *Lq*Þð Þ*<sup>r</sup>* , *Nq*Þð Þ*<sup>r</sup>* ,*W<sup>m</sup> <sup>q</sup>*Þð Þ*<sup>r</sup>* and *WN<sup>m</sup> <sup>q</sup>*Þð Þ*<sup>r</sup>* , respectively. Along with *WN<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> , consider the following space of functions *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> equipped with the norm

$$||f||\_{N\_q^w(\Omega)} = \sum\_{|p| \le m} d\_{\Omega}^{|p| - \frac{n}{q}} ||\partial^p f||\_{L\_{q)(\Omega)}},\tag{17}$$

where *<sup>d</sup>*<sup>Ω</sup> <sup>¼</sup> *diam* <sup>Ω</sup>, and we will assume *<sup>N</sup>*<sup>0</sup> *<sup>q</sup>*Þð Þ¼ <sup>Ω</sup> *Nq*Þð Þ <sup>Ω</sup> . The closure of *<sup>C</sup>*<sup>∞</sup> <sup>0</sup> ð Þ Ω in *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> (*N<sup>m</sup> <sup>q</sup>* ð Þ <sup>Ω</sup> ) we will denote by <sup>∘</sup>*N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> (∘*N<sup>m</sup> <sup>q</sup>* ð Þ Ω ).

#### **3. Main lemma**

#### **3.1 Solvability in the small**

Introduce the following

**Definition 1.7** We will say that the operator *L* has the property *Px*<sup>0</sup> Þ if its coefficients satisfy the conditions: i) *ap* ∈*L*∞ð Þ *Br*ð Þ *x*<sup>0</sup> , ∀j j *p* ≤ *m*, for some *r*>0; ii) ∃*r*> 0 : for j j *p* ¼ *m* the coefficient *ap*ð Þ� *coincides a.e. in Br*ð Þ *x*<sup>0</sup> *with some function bounded and continuous at the point x*0.

It is absolutely clear that if *ap* ∈*C*ð Þ Ω , ∀j j *p* ≤ *m*, then *L* has the property *Px*<sup>0</sup> Þ for ∀*x*<sup>0</sup> ∈ Ω.

In establishing the interior Schauder-type estimate for grand-Sobolev spaces *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> the following Main Lemma, proved in [18] plays a key role.

**Main Lemma.** *Let the m-th order elliptic operator L have the property Px*<sup>0</sup> Þ *at the point x*0*. Let φ* ∈ *N<sup>m</sup> <sup>q</sup>*Þð Þ *Br*ð Þ *<sup>x</sup>*<sup>0</sup> *and <sup>φ</sup> vanish in a neighborhood of x*j j � *<sup>x</sup>*<sup>0</sup> <sup>¼</sup> *r. Then for q*><sup>1</sup> *it holds*

$$||T\_{\mathbf{x}\_0}\varrho||\_{N^m\_{q\cdot}(B\_r(\mathbf{x}\_0))} \le \sigma(r)||\varrho||\_{N^m\_{q\cdot}(B\_r(\mathbf{x}\_0))},\tag{18}$$

where the function *σ*ð Þ!*r* 0,*r* ! 0, depends only on the ellipticity constant *Lx*<sup>0</sup> , on the coefficients of *L* and their moduli of continuity.

Let us consider the *m*-th order elliptic operator *L* with the coefficients *ap*ð Þ *x* defined by (1), and the corresponding operator *Tx*<sup>0</sup> defined by (5). Using Main Lemma, it is proved the following local existence theorem.

**Theorem 1.8** Let *L* be an *m*-th order elliptic operator which has the property *Px*<sup>0</sup> Þ at some point *x*<sup>0</sup> ∈ *Ω* and *f* ∈ *Gq*Þð Þ *Ω* , 1<*q* < þ ∞. Then, for sufficiently small *r*, there exists a solution of the equation *Lu* ¼ *f* belonging to the class *Nq*Þð Þ *Br*ð Þ *x*<sup>0</sup> *.*

#### **4. Interior Schauder type estimates**

Let *<sup>ω</sup>*ð Þ� be an infinitely differentiable function on 0, 1 ½ � such that for 0≤*t*<sup>&</sup>lt; <sup>1</sup> 3 , *<sup>ω</sup>*ðÞ� *<sup>t</sup>* 1 and for <sup>2</sup> <sup>3</sup> <*t*≤ 1, *ω*ðÞ� *t* 0. For 0< *R*<sup>1</sup> <*R*<sup>2</sup> we put

$$\xi(\mathbf{x}) = \begin{cases} \mathbf{1}, |\mathbf{x}| \le R\_1, \\ \alpha \left( \frac{|\mathbf{x}| - R\_1}{R\_2 - R\_1} \right), R\_1 < |\mathbf{x}| \le R\_2. \end{cases} \tag{19}$$

Regarding this function it holds the following.

**Lemma 1.9** *There is a constant C*>0 *depending only on R*<sup>2</sup> *and ω*ð Þ� *, such that for* ∀*R*<sup>1</sup> : 0<*R*<sup>1</sup> < *R*2*, there is*

$$||\xi||\_{C^{m}(R\_{2})} \leq C \left(1 - \frac{R\_{1}}{R\_{2}}\right)^{-m} . \tag{20}$$

Accept the following property with respect to the domain Ω.

*Property α*Þ*.We say that the domain Ω admits the continuation of functions of the space Nk <sup>q</sup>*Þð Þ *<sup>Ω</sup> if there exists a domain <sup>Ω</sup>*<sup>0</sup> <sup>⊃</sup> *<sup>Ω</sup> and a linear mapping <sup>θ</sup> of the space N<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> into Nk <sup>q</sup>*<sup>Þ</sup> *<sup>Ω</sup>*<sup>0</sup> ð Þ *such that*

$$\begin{aligned} \theta u &= u \text{ in } \Omega, \\ ||\theta u||\_{N^k\_{(q)}(\Omega^\circ)} &\le \text{const} \, ||u||\_{N^k\_{(q)}(\Omega)}, \end{aligned} \tag{21}$$

holds.

So, the following lemma is true.

**Lemma 1.10** *Let the domain <sup>Ω</sup> have the Property <sup>α</sup>*<sup>Þ</sup> *with respect to space N<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> . Then* ∃*C*>0 *depending only on n*, *q and on a constant from* (21)*, which holds*

$$||\varrho||\_{N\_{q}^{k}(\Omega)} \leq \varepsilon ||\varrho||\_{N\_{q}^{k+1}(\Omega)} + \mathsf{C}\varepsilon^{-k}||\varrho||\_{L\_{q}(\Omega)},\tag{22}$$

for ∀*k* ¼ 1, *m* � 1 and ∀*ε*> 0.

The main result of this section is the following Schauder type estimate.

**Theorem 1.11** *Let the coefficients of m-th order elliptic operator L satisfy the following conditions: i) ap*ð Þ� ∈*C Ω ,* <sup>∀</sup>*<sup>p</sup>* : j j *<sup>p</sup>* <sup>¼</sup> *m; ii) ap*ð Þ� <sup>∈</sup> *<sup>L</sup>*∞ð Þ *<sup>Ω</sup> ,* <sup>∀</sup>*<sup>p</sup>* : j j *<sup>p</sup>* <sup>&</sup>lt; *m; where <sup>Ω</sup>* <sup>⊂</sup>*Rn*� *bounded domain with boundary <sup>∂</sup>Ω. Let <sup>Ω</sup>*<sup>0</sup> <sup>⊂</sup> *<sup>Ω</sup> be an arbitrary compact. Then for* ∀*u*∈*WG<sup>m</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> ,* <sup>1</sup><*<sup>q</sup>* <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>*, the following a priori estimate holds*

$$||u||\_{N\_{q}^{m}(\Omega\_{0})} \leq C \Big( ||Lu||\_{L\_{q}(\Omega)} + ||u||\_{L\_{q}(\Omega)} \Big),\tag{23}$$

where the constant *C* depends only on the ellipticity constant *m*, Ω, Ω<sup>0</sup> of *L*, on the coefficients of the operator *L*.

#### **5. Extension of functions from** *N<sup>m</sup> q*Þ ð Þ **Ω Compactness**

Consider the question of the possibility of extension of a function *f* from class *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> to a wider class *<sup>N</sup><sup>m</sup> <sup>q</sup>*<sup>Þ</sup> <sup>Ω</sup><sup>0</sup> ð Þ with <sup>Ω</sup><sup>0</sup> ⊃Ω. Following the classical case (see, monograph [29]), first consider the case when Ω<sup>0</sup> is a cube with an edge 2*a*> 0 : *Ka* ¼ *yi* <*a*, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>* , and <sup>Ω</sup> is a parallelepiped *<sup>K</sup>*<sup>þ</sup> *<sup>a</sup>* <sup>¼</sup> *Ka*<sup>⋂</sup> *yn* <sup>&</sup>gt; <sup>0</sup> .

The following lemma is true.

**Lemma 1.12** *For* ∀*f* ∈ *WN<sup>k</sup> <sup>q</sup>*<sup>Þ</sup> *<sup>K</sup>*<sup>þ</sup> *a there exists an extension F* ∈*WN<sup>k</sup> <sup>q</sup>*Þð Þ *Ka and, in addition, inequality*

$$\|\|F\|\|\_{N\_{q}^{m}(\mathbb{K}\_{a})} \leq C \|f\|\|\_{N\_{q}^{m}(\mathbb{K}\_{a}^{+})},\tag{24}$$

holds.

It is completely analogous to the monograph [29, p. 129], it is proved the following

**Lemma 1.13** Let *f* ∈*WN<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* and for <sup>∀</sup>*ξ*<sup>∈</sup> *<sup>∂</sup><sup>Ω</sup>* there exists a function *<sup>F</sup>ξ*ð Þ *<sup>x</sup> , defined in a ball Br*ð Þ*ξ* of some radius *r* ¼ *r*ð Þ*ξ* > 0, such that *Fξ*ð Þ¼ *x f x*ð Þ, ∀*x*∈ *Ω*∩*Br*ð Þ*ξ and F<sup>ξ</sup>* ∈ *WN<sup>k</sup> <sup>q</sup>*Þð Þ *Br*ð Þ*<sup>ξ</sup> .* Besides

$$\left\|\left|F\_{\xi}\right|\right\|\_{\mathcal{W}\mathcal{N}^{k}\_{q\mid}(B\_{r}(\xi))} \leq \mathcal{C} \left\|f\right\|\vert\_{\mathcal{W}\mathcal{N}^{k}\_{q\mid}(\mathfrak{Q})},\tag{25}$$

is true, where *C*>0 is a constant independent of *f*. Then, for any *ρ*>0, the function *f* has an extension *F* to the domain Ω*<sup>ρ</sup>* ¼ ⋃ *x*∈ Ω *Bρ*ð Þ *x* with the properties *F* ∈*WN<sup>k</sup> <sup>q</sup>*<sup>Þ</sup> Ω*<sup>ρ</sup>* , *F x*ð Þ¼ 0, <sup>∀</sup>*x*<sup>∈</sup> <sup>Ω</sup>*ρ*nΩ*<sup>ρ</sup>=*2: and the inequality

$$\|\|F\|\|\_{\text{WN}^{\mathfrak{k}}\_{q\mathfrak{l}}\left(\Omega\_{\mathfrak{p}}\right)} \leq C \|f\|\|\_{\text{WN}^{\mathfrak{k}}\_{q\mathfrak{l}}\left(\Omega\right)},\tag{26}$$

holds, where the constant *C*>0, is dependent only on domain Ω and *ρ*.

Using Main Lemma and Lemmas 1.12, 1.13, similarly to [29, p. 130] it is proved the following extension.

**Theorem 1.14** *Let Ω, Ω*<sup>0</sup> *be bounded domains in Rn, Ω* ⊂ *Ω*<sup>0</sup> *and ∂Ω* ∈*C*ð Þ *<sup>m</sup> . Then for* ∀*f* ∈ *N<sup>m</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> there exists a finite extension F* <sup>∈</sup> *<sup>N</sup><sup>m</sup> <sup>q</sup>*<sup>Þ</sup> *<sup>Ω</sup>*<sup>0</sup> ð Þ *in <sup>Ω</sup>*<sup>0</sup> *and the following estimate*

$$\|F\|\_{N\_{q}^{m}(\Omega')} \leq C \|f\|\_{N\_{q}^{m}(\Omega)},\tag{27}$$

is valid, where *C*> 0 is a constant independent of *f*.

Consider the compactness question of the family in *Nq*Þð Þ Ω . The following theorem is true.

**Theorem 1.15** Let *Ω* ⊂ *Rn* be a bounded domain with a boundary *∂Ω* ∈*C*ð Þ<sup>1</sup> *. Then a set, bounded in N*<sup>1</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup> , is compact in Nq*Þð Þ *<sup>Ω</sup> .*

Analogously to Theorem 1.15, the following theorem is also proved.

**Theorem 1.16** Let *Ω* ⊂*Rn* be a bounded domain with a boundary *∂Ω* ∈*C*ð Þ*<sup>k</sup>* . If the set of functions is bounded in *N<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* , then the set of their traces on ð Þ *<sup>n</sup>* � <sup>1</sup> dimensional surface *Γ* ⊂ *Ω* from the class *C*ð Þ*<sup>k</sup>* is compact in *W<sup>r</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* , <sup>∀</sup>*<sup>r</sup>* <sup>¼</sup> 0, *<sup>k</sup>* � 1.

#### **6. Trace of functions from the grand-Sobolev space** *N<sup>m</sup> <sup>q</sup>*Þð Þ **Ω**

In this section, we will define a concept of the trace for functions from the grand-Sobolev space *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> on an ð Þ *<sup>n</sup>* � <sup>1</sup> -dimensional differentiable surface.

Based on the embedding *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> <sup>⊂</sup> *<sup>N</sup>*<sup>1</sup> *<sup>q</sup>*Þð Þ Ω , ∀*m* ≥2, it is sufficient to define this concept regarding the functions from *N*<sup>1</sup> *<sup>q</sup>*Þð Þ <sup>Ω</sup> . So, assume *<sup>S</sup>*<sup>⊂</sup> <sup>Ω</sup>: *<sup>S</sup>*∈*C*ð Þ<sup>1</sup> is some ð Þ *<sup>n</sup>* � <sup>1</sup> dimensional surface. Let *x*<sup>0</sup> ∈ *S* be an arbitrary point. Then it is obvious that there exists a sufficiently small neighborhood of this point *Sx*<sup>0</sup> ⊂ *S*, such that uniquely projected onto some domain *<sup>D</sup>* of the plane *xn* <sup>¼</sup> 0 in *<sup>R</sup><sup>n</sup>* and it has the equation

$$\mathfrak{x}\_{\mathfrak{n}} = \mathfrak{q}(\mathfrak{x}') \in \mathbb{C}^{(1)}(\bar{D}), \mathfrak{x}' = (\mathfrak{x}\_1, \dots, \mathfrak{x}\_{n-1}) \in D. \tag{28}$$

Ω is bounded domain and we will consider that it is placed inside a cube <sup>0</sup><sup>&</sup>lt; *xi* <sup>&</sup>lt;*a*, *<sup>i</sup>* <sup>¼</sup> 1, *<sup>n</sup>* � �, with an edge *<sup>a</sup>*>0. Let *<sup>f</sup>* ∈ ∘*C*<sup>∞</sup>ð Þ <sup>Ω</sup> be some finite function in <sup>Ω</sup>. For ∀ *x*<sup>0</sup> ; *φ x*<sup>0</sup> ð Þ ð Þ ∈*Sx*<sup>0</sup> we have

$$f(\mathbf{x})/\_{\mathbb{S}\_{\mathbf{x}\_0}} = f(\mathbf{x}'; \boldsymbol{\varrho}(\mathbf{x}')) = \int\_0^{\boldsymbol{\varrho}(\mathbf{x}')} \frac{\partial f(\mathbf{x}'; \boldsymbol{\xi}\_n)}{\partial \boldsymbol{\xi}\_n} d\boldsymbol{\xi}\_n. \tag{29}$$

Let *<sup>ε</sup>*∈ð Þ� 0, *<sup>q</sup>* � <sup>1</sup> be an arbitrary number, *<sup>q</sup><sup>ε</sup>* <sup>¼</sup> *<sup>q</sup>* � *<sup>ε</sup>* and <sup>1</sup> *qε* <sup>þ</sup> <sup>1</sup> *q*0 *<sup>ε</sup>* ¼ 1. Applying Hölder's inequality from (29), we obtain

$$\left| f \right/\_{S\_{\pi\_0}} \right|^{q\_\epsilon} \le |\rho(\mathbf{x'})|^{\frac{q\_\epsilon}{q\_\epsilon}} \int\_0^{\rho(\mathbf{x'})} \left| \frac{\partial f(\mathbf{x'}; \boldsymbol{\xi}\_n)}{\partial \boldsymbol{\xi}\_n} \right|^{q\_\epsilon} d\boldsymbol{\xi}\_n \le a^{\frac{q\_\epsilon}{q\_\epsilon}} \int\_0^{\rho(\mathbf{x'})} \left| \frac{\partial f(\mathbf{x'}; \boldsymbol{\xi}\_n)}{\partial \boldsymbol{\xi}\_n} \right|^{q\_\epsilon} d\boldsymbol{\xi}\_n. \tag{30}$$

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

Let *C* ¼ max f g *a*; 1 . Consequently

$$\varepsilon \left| f \left/ \_{S\_{x\_0}} \right| ^{q\_\times} \leq \mathcal{C}^{q\_\times} \varepsilon \int\_0^{\rho(\mathbf{x}^\prime)} \left| \frac{\partial f(\mathbf{x}^\prime; \xi\_n)}{\partial \xi\_n} \right| ^{q\_\times} d\xi\_n \right. \tag{31}$$

where *C* is a constant independent of *f* and *ε*. Multiplying by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ *φ*<sup>2</sup> *<sup>x</sup>*<sup>1</sup> þ … þ *φ*<sup>2</sup> *xn*�<sup>1</sup> q and integrating over *D* we obtain

$$\left\| \left| \boldsymbol{\varepsilon}^{\mathsf{T}}\_{\boldsymbol{\mathsf{f}}} \boldsymbol{f} \right| \right\|\_{L\_{q\_{\boldsymbol{\mathsf{f}}}}\left(\mathbb{S}\_{\boldsymbol{\mathsf{x}}\_{0}}\right)}^{q\_{\boldsymbol{\mathsf{f}}}} \leq \mathsf{C} \left\| \left| \boldsymbol{\varepsilon}^{\mathsf{T}} \frac{\partial \boldsymbol{f}}{\partial \mathsf{x}\_{\boldsymbol{\mathsf{n}}}} \right| \right\|\_{L\_{q\_{\boldsymbol{\mathsf{x}}}}\left(\mathfrak{Q}\right)}^{q\_{\boldsymbol{\mathsf{f}}}}.\tag{32}$$

Since the surface *S* can be covered by a finite number of surfaces of type *Sx*<sup>0</sup> , then summing the corresponding inequalities (from (32)) we establish

$$\left\| \left| e^{\frac{\Delta}{q\_r}} f \right| \right\|\_{L\_{q\_r}(S)}^{q\_r} \le C \left\| \left| e^{\frac{\Delta}{q\_r}} \frac{\partial f}{\partial \mathbf{x}\_n} \right| \right\|\_{L\_{q\_r}(\mathfrak{Q})}^{q\_r},\tag{33}$$

where *C*> 0 is a constant independent of *f* and *ε*. This immediately implies

$$\left\| \left| \varepsilon^{\frac{1}{q\_r}} f \right| \right\|\_{L\_{q\_r}(\mathbb{S})} \leq C \sum\_{|p|=1} d\_{\Omega} \left\| \left| \varepsilon^{\frac{1}{q\_r}} \partial^p f \right| \right\|\_{L\_{q\_r}(\Omega)} \leq C \left\| \left| \varepsilon^{\frac{1}{q\_r}} \partial^p f \right| \right\|\_{N\_{q\_r}^4(\Omega)}.\tag{34}$$

Taking first sup 0 <*ε*<*q*�1 on the right and then the same sup on the left, from this 0

estimate for ∀*f* ∈*C* ∞ð Þ Ω , we have

$$\|f\|\_{L\_q(\mathcal{S})} \le \mathcal{C} \|f\|\_{N^1\_{q\mathcal{I}}(\mathfrak{Q})}.\tag{35}$$

If ∂Ω ∈*C*ð Þ<sup>1</sup> , then Theorem 1.14 implies that the inequality (35) holds for ∀*f* ∈*C*ð Þ<sup>1</sup> Ω � �.

Let *f* ∈ *N*<sup>1</sup> *<sup>q</sup>*Þð Þ Ω be an arbitrary function. Then ∃ *f <sup>n</sup>* � �⊂*C*<sup>∞</sup> Ω � �:

$$\left\| \left. f\_n - f \right| \right\|\_{N^1\_{q/}(\Omega)} \to 0, n \to \infty. \tag{36}$$

It follows directly from (35) that the sequence *f <sup>n</sup>* � � is fundamental in *Lq*Þð Þ*<sup>S</sup>* :

$$\left\| \left\| f\_n - f\_m \right\| \right\|\_{L\_q(\mathbb{S})} \to 0, n, m \to \infty. \tag{37}$$

From the completeness of *Lq*Þð Þ*S* it follows that ∃ *f <sup>S</sup>* ∈ *Lq*Þð Þ*S* :

$$\left\| \left\| f\_n - f\_S \right\| \right\|\_{L\_\eta(S)} \to \mathbf{0}, n \to \infty. \tag{38}$$

Similarly the classical case, it is proved that *f <sup>S</sup>* does not depend on the choice of the sequence *f <sup>n</sup>* � �.

*f <sup>S</sup>* is called the trace of the function *f* ∈ *N*<sup>1</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>* on *<sup>S</sup>* and we will denote it by the operator *Γ<sup>S</sup>* : *Γf* ¼ *f =S.*

Based on the concept of Γ*S*, we define the following linear space

$$N\_{q\rangle}^1(\mathbb{S}) = N\_{q\rangle}^1(\mathbb{\Omega})/\_{\mathbb{S}} = \left\{ f \in L\_{q\rangle}(\mathbb{S}) : \exists u \in N\_{q\rangle}^1(\mathbb{\Omega}) \Rightarrow f = \Gamma\_{\mathbb{S}}u = u/\_{\mathbb{S}} \right\}.\tag{39}$$

For the case of *S* ¼ ∂Ω, the operator Γ*<sup>S</sup>* will be simply denoted by Γ : Γ<sup>∂</sup><sup>Ω</sup> ¼ Γ*:* The following lemma is true.

**Lemma 1.17** *Let Ω* ⊂*R<sup>n</sup> be an bounded domain and ∂Ω* ∈*C*ð Þ<sup>1</sup> *. Then the linear spaces* F1 *<sup>q</sup>*<sup>Þ</sup> *and N*<sup>1</sup> *<sup>q</sup>*Þð Þ *<sup>∂</sup><sup>Ω</sup> are isomorphic, where*

$$N\_{q/}^{1}(\partial\Omega) \equiv N\_{q/}^{1}(\Omega)/\_{d\Omega} = \left\{ f \in L\_{q}(\partial\Omega) : \exists u \in N\_{q/}^{1}(\Omega) \Rightarrow f = \Gamma u = u/\_{d\Omega} \right\}.\tag{40}$$

Based on this lemma, we define the norm in *N*<sup>1</sup> *<sup>q</sup>*Þð Þ ∂Ω

$$\|f\|\_{N^1\_{q\rangle}(\partial\Omega)} = \left| \|\Gamma^{-1}f\|\right|\_{\mathcal{F}^1\_{q\rangle}}, \forall f \in \mathcal{N}^1\_{q\rangle}(\partial\Omega). \tag{41}$$

Since F<sup>1</sup> *<sup>q</sup>*<sup>Þ</sup> is a Banach space with respect to the norm k k� <sup>F</sup><sup>1</sup> *q*Þ , then this lemma immediately implies that *N*<sup>1</sup> *<sup>q</sup>*Þð Þ ∂Ω is also Banach with respect to the norm (41).

The space *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>∂</sup><sup>Ω</sup> is defined similarly and the corresponding lemma is true for the spaces F *<sup>m</sup> <sup>q</sup>*Þ, where

$$N\_{q/}^{m}(\partial\Omega) \equiv N\_{q/}^{m}(\Omega)/\_{\partial\Omega} = \left\{ f \in L\_{q}(\partial\Omega) : \exists u \in N\_{q/}^{m}(\Omega) \Rightarrow f = \Gamma u = u/\_{\partial\Omega} \right\}.\tag{42}$$

The following theorem is true (regarding the proof see [30]).

**Theorem 1.18** Let *Ω* ⊂*R<sup>n</sup>* be a bounded domain with a boundary *∂Ω* ∈*C*ð Þ *<sup>m</sup> . If the set of functions is bounded in N<sup>m</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* , *<sup>m</sup>* <sup>≥</sup> 1, then the set of their traces on the ð Þ *<sup>n</sup>* � <sup>1</sup> dimensional surface *<sup>S</sup>*<sup>⊂</sup> *<sup>Ω</sup> from the class C*ð Þ *<sup>m</sup> is compact in Lq*Þð Þ*<sup>S</sup> .*

#### **7. Schauder-type estimate up to the boundary**

Using the results obtained in the previous sections, it is established a Schaudertype estimate up to the boundary for a second-order elliptic operator with nonsmooth coefficients. The following theorem is true.

**Theorem 1.19** *Let Ω* ⊂*R<sup>n</sup> be a bounded domain with a boundary ∂Ω* ∈*C*ð Þ<sup>2</sup> *and L be a second-order elliptic operator (i.e. m* <sup>¼</sup> <sup>2</sup>*) with coefficients ap* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup>*� , <sup>∀</sup>*<sup>p</sup>* : j j *<sup>p</sup>* <sup>¼</sup> *m and ap* <sup>∈</sup>*L*∞ð Þ *<sup>Ω</sup>* , <sup>∀</sup>*<sup>p</sup>* : j j *<sup>p</sup>* <sup>&</sup>lt; *m. Then for* <sup>∀</sup>*<sup>u</sup>* <sup>∈</sup> *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup> the following estimate*

$$||\mathfrak{u}||\_{N\_{q}^{2}(\mathfrak{\Omega})} \leq \mathbf{C} \Big( ||Lu||\_{N\_{q}(\mathfrak{\Omega})} + ||u||\_{N\_{q}^{2}(\mathfrak{\partial}\mathfrak{\Omega})} + ||u||\_{N\_{q}(\mathfrak{\Omega})} \Big),\tag{43}$$

holds true, where *C*> 0 is a constant independent of *u*, but depends on the norms of the coefficients of *L* in *L*∞ð Þ Ω .

#### **8. Solvability of the Dirichlet problem for a second-order elliptic operator**

Let us apply the estimates established in the previous sections to the solvability question (in the strong sense) of the Dirichlet problem for a second-order elliptic type equation in classes *N*<sup>2</sup> *<sup>q</sup>*Þð Þ <sup>Ω</sup> . So, let <sup>Ω</sup> <sup>⊂</sup> *Rn* be a domain with a boundary <sup>Ω</sup> <sup>∈</sup>*C*ð Þ<sup>2</sup> . Assume that *<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ <sup>Ω</sup> is a given function and *aij* <sup>∈</sup>*C*ð Þ <sup>Ω</sup>� ; *ai*; *<sup>a</sup>*<sup>∈</sup> *<sup>L</sup>*∞ð Þ <sup>Ω</sup> *<sup>i</sup>*;*<sup>j</sup>* <sup>¼</sup> 1, *<sup>n</sup>*. Consider the equation

$$Lu = \sum\_{i,j=1}^{n} a\_{\vec{\eta}}(\mathbf{x}) \frac{\partial^2 u}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + \sum\_{i=1}^{n} a\_i(\mathbf{x}) \frac{\partial u}{\partial \mathbf{x}\_i} + a(\mathbf{x}) u = f(\mathbf{x}), \mathbf{x} \in \Omega. \tag{44}$$

In the sequel we will assume that the following uniformly ellipticity condition holds a.e. in Ω

$$\left|\nu|\xi|^2 \le \sum\_{i,j=1}^n a\_{ij}(\varkappa)\xi\_i\xi\_j \le \nu^{-1}|\xi|^2, \forall \xi \in R^n,\tag{45}$$

where *ν*∈ð � 0, 1 is some constant.

Under the solution of the Eq. (44) we mean a function *u* ∈ *N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω for which equality (44) holds a.e. *x*∈ Ω. Let *φ*∈ *N*<sup>2</sup> *<sup>q</sup>*Þð Þ ∂Ω be a given function. Let us define the boundary condition

$$
\Gamma \mathfrak{u} = \mathfrak{u}/\_{\mathfrak{A}\mathfrak{A}} = \mathfrak{q} \tag{46}
$$

where Γ : *N*<sup>2</sup> *<sup>q</sup>*Þð Þ! <sup>Ω</sup> *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ� ∂ Ω is a trace operator.

We will say that the domain Ω has a property *δ<sup>q</sup>* � � is class ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω , if the Dirichlet problem (46) is correctly solvable for the Poisson equation, i.e. the problem

$$\begin{cases} \Delta u = f, \text{in } \Omega, \\ \Gamma u = 0, on \text{ } \partial \Omega, \end{cases} \tag{47}$$

has a unique solution for <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ <sup>Ω</sup> in class <sup>∘</sup>*N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω .

In order to solve the problem (44), (46) we apply the parameter continuation method (see e.g. [28, p. 247]).

Furthermore, assume that the operator *L* satisfies the following inequality

$$\|u\|\_{L\_q(\mathfrak{Q})} \le C \|Lu\|\_{L\_q(\mathfrak{Q})}, \forall u \in \circ \mathcal{N}\_q^2(\mathfrak{Q}),\tag{48}$$

where the constant *C* depends only on the ellipticity constants of the operator *L*, on the sup norms of the coefficients *L*, on domain Ω and is independent of the function *u*∈ ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω .

We will say that the operator L has property (A) if an inequality (48) holds for an operator L.

The question of whether inequality (48) holds (i.e. property (A)) we will consider later.

Thus, the following main theorem is true.

**Theorem 1.20** *Let Ω* ⊂*R<sup>n</sup> be a bounded domain with a boundary ∂Ω* ∈*C*ð Þ<sup>2</sup> *and L is a second-order elliptic differential operator defined by* expression (44) *with* coefficients *aij* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup>*� *; ai; a*∈*L*∞ð Þ *<sup>Ω</sup> , i*; *<sup>j</sup>* <sup>¼</sup> 1, *n. Assume that the domain <sup>Ω</sup> has property <sup>Δ</sup>q*<sup>Þ</sup> *in class* ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup> and the operator L has property (A). Then the equation Lu* <sup>¼</sup> *f is uniquely solvable for* <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ *<sup>Ω</sup> in class* <sup>∘</sup>*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup> , i.e.* <sup>∘</sup>*<sup>L</sup>* : *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ\$ *<sup>Ω</sup> Nq*Þð Þ *<sup>Ω</sup> is an isomorphism and it is obvious that the estimate*

$$\|\|u\|\|\_{N\_{q}^{2}(\Omega)} \leq C \|f\|\|\_{N\_{q}(\Omega)}, \forall f \in \mathcal{N}\_{q}(\Omega),\tag{49}$$

holds true, where *C*> 0 is a constant independent of *f*.

Now consider a homogeneous equation *Lu* ¼ 0 in Ω with a nonhomogeneous boundary condition <sup>Γ</sup>*<sup>u</sup>* <sup>¼</sup> *<sup>u</sup>=*∂<sup>Ω</sup> <sup>¼</sup> *<sup>φ</sup>*, where *<sup>φ</sup>*<sup>∈</sup> *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ� Ω is given function. From the results of Section 6 it follows that ∃Φ ∈ *N*<sup>2</sup> *<sup>q</sup>*Þð Þ <sup>Ω</sup> : ΓΦ <sup>¼</sup> *<sup>φ</sup>:* Suppose *<sup>υ</sup>* <sup>¼</sup> *<sup>u</sup>* � <sup>Φ</sup> and let *f* ¼ �*L*Φ. It is clear that Γ*υ* ¼ *υ=*∂<sup>Ω</sup> ¼ 0 and *Lυ* ¼ *f* in Ω. If *aij*; *ai*; *a*∈*L*∞ð Þ Ω , *i*; *j* ¼ 1, *n*, then, as follows from Proposition 1.6 that *f* ∈ *Nq*Þð Þ Ω . Therefore, we can apply the Theorem 1.20 to the problem

$$\begin{cases} \text{L}\boldsymbol{\nu} = \boldsymbol{f}, \boldsymbol{a}.e.\boldsymbol{e}.\text{ in }\boldsymbol{\Omega},\\ \boldsymbol{\nu}/\_{\partial\Omega} = \mathbf{0}. \end{cases} \tag{50}$$

If all the conditions of Theorem 1.20 are satisfied, then this problem is uniquely solvable in class *N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω and for the solution it is valid the following estimate

$$\|\|\nu\|\|\_{N\_{q\cdot}^{2}(\Omega)} \leq C \|f\|\|\_{N\_{q\cdot}(\Omega)},\tag{51}$$

where *C*> 0 is a constant independent of *f*. It is quite obvious that then the problem

$$\begin{cases} Lu = 0, a.e.in\ \Omega, \\ u/\_{\Gamma} = \wp, \end{cases} \tag{52}$$

is also uniquely solvable in *N*<sup>2</sup> *<sup>q</sup>*Þð Þ Ω .

Taking into account expression (41) for the norm in *N*<sup>2</sup> *<sup>q</sup>*Þð Þ ∂Ω , we obtain

$$\|\|u\|\|\_{N\_{q\cdot}^{2}(\Omega)} \leq C \|\|\rho\|\|\_{N\_{q\cdot}^{2}(\partial\Omega)}.\tag{53}$$

Consider a nonhomogeneous equation with a nonhomogeneous boundary condition

$$\begin{cases} \mathcal{L}u = f \ a.e.in \ \Omega, \\ u /\_{\partial \Omega} = \emptyset, \end{cases} \tag{54}$$

where *<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ <sup>Ω</sup> and *<sup>φ</sup>*<sup>∈</sup> *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ� ∂Ω are given functions. Representing the function *u* in the form *u* ¼ *v* þ *w*, where

$$\begin{cases} Lv = f, \\ v/\_{\partial \Omega} = 0, \int \omega/\_{\partial \Omega} = \rho, \end{cases} \tag{55}$$

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

from Theorem 1.20 and taking into account estimate (53), we arrive at the following conclusion.

**Theorem 1.21** *Let the domain Ω and the operator L satisfy all the conditions of Theorem 1.20. Then for* <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ *<sup>Ω</sup> and* <sup>∀</sup>*<sup>φ</sup>* <sup>∈</sup> *<sup>N</sup>*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>∂</sup><sup>Ω</sup> the problem* (54) *is uniquely solvable in the space N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>∂</sup><sup>Ω</sup> and regarding the solution the estimate*

$$\|\|u\|\|\_{N\_q^2(\Omega)} \le C \left( \|f\|\|\_{N\_q(\Omega)} + \|\|\rho\|\|\_{L\_q(\partial\Omega)} \right),\tag{56}$$

is fulfilled, where *C*> 0 is a constant independent of *f* and *φ*.

#### **9. Some properties of a Riesz potential**

For obtaining main results we need some properties of a Riesz potential and embedding theorems regarding the spaces *N<sup>m</sup>* <sup>q</sup>Þ. In this section, we will give some properties of an integral operator with a weak singularity. These properties are used to study the properties of functions from class *W<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> . Let us remember the Sobolev integral identity

$$u(\mathbf{x}) = \sum\_{|a|=0}^{k-1} \mathbf{x}^a \Big|\_{\Omega} b\_a(\mathbf{y}) u(\mathbf{y}) d\mathbf{y} + \sum\_{|a|=k} \int\_{\Omega} \frac{A\_a(\mathbf{x}; \mathbf{y})}{r^{n-k}} \partial^a u(\mathbf{y}) d\mathbf{y}, \forall u \in \mathbf{C}^k(\bar{\Omega}), \tag{57}$$

where *<sup>b</sup><sup>α</sup>* <sup>∈</sup>*C*ð Þ <sup>Ω</sup>� , *<sup>A</sup>*<sup>∝</sup> <sup>∈</sup> *<sup>L</sup>*∞ð Þ <sup>Ω</sup> � <sup>Ω</sup> (generally speaking, for *<sup>x</sup>* 6¼ *<sup>y</sup>*: *<sup>A</sup>*∝ð Þ *<sup>x</sup>*; *<sup>y</sup>* is infinitely differentiable). In establishing many properties of a function from Sobolev classes the representation (57) plays a key role. In accordance with (57) consider the integral operator (Riesz potential).

$$(K\rho)(\mathfrak{x}) = V(\mathfrak{x}) = \int\_{\mathfrak{A}} \frac{A(\mathfrak{x}, \mathfrak{y})}{r^l} \rho(\mathfrak{y}) d\mathfrak{y},\tag{58}$$

where *<sup>r</sup>* <sup>¼</sup> j j *<sup>y</sup>* � *<sup>x</sup>* ; *<sup>x</sup>*<sup>∈</sup> <sup>Ω</sup> <sup>⊂</sup>*R<sup>n</sup>* is a bounded domain, 0 <sup>≤</sup>*λ*<sup>&</sup>lt; *<sup>n</sup>*; *<sup>A</sup>* <sup>∈</sup>*L*∞ð Þ <sup>Ω</sup> � <sup>Ω</sup> . The following theorem is true.

**Theorem 1.22** *Let* <sup>Ω</sup> <sup>⊂</sup>*R<sup>n</sup>* be a bounded domain, *<sup>ρ</sup>*∈*Lq*Þð Þ <sup>Ω</sup> , *<sup>A</sup>* <sup>∈</sup>*C*ð Þ <sup>Ω</sup>� � <sup>Ω</sup>� and *<sup>λ</sup>q*<sup>0</sup> <sup>&</sup>lt;*n*. Then operator (58) acts compactly from *<sup>L</sup>*qÞð Þ <sup>Ω</sup> to *<sup>C</sup>*ð Þ <sup>Ω</sup>� *.* It is true the following classical analogue

**Theorem 1.23** Let *λq*<sup>0</sup> ≥*n* and an integer *s* satisfy *n* � ð Þ *n* � *λ* <*s*≤*n*. Then the integral (58) defines a function that, on any intersection *Ω<sup>s</sup>* of the set *Ω* by a plane of dimension *s*, is defined almost everywhere in the sense of the Lebesgue measure in *R<sup>s</sup>* . The operator *K* defined by formula (58) is bounded as an operator from *Lq*Þð Þ *<sup>Ω</sup> to Lr*ð Þ *<sup>Ω</sup><sup>s</sup> (also from Lq*Þð Þ *<sup>Ω</sup> to Lr*Þð Þ *<sup>Ω</sup><sup>s</sup>* ), for <sup>∀</sup>*r*: 1<sup>&</sup>lt; *<sup>r</sup>*<sup>&</sup>lt; *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *sq <sup>n</sup>*�ð Þ *<sup>n</sup>*�*<sup>λ</sup> q.* It is valid the following

**Theorem 1.24** If *λq*<sup>0</sup> ≥*n*, then the operator *K*, defined by expression (58), acts compactly from *Lq*Þð Þ *Ω to Lr*ð Þ *Ω (also from Lq*Þð Þ *Ω to Lr*Þð Þ *Ω* ), for ∀*r* : 1< *r*<*r*<sup>0</sup> ¼ *nq <sup>n</sup>*�ð Þ *<sup>n</sup>*�*<sup>λ</sup> q.*

#### **10. Embedding theorems**

To obtain Sobolev-type embedding theorems in spaces *W<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> we will use the results obtained in the previous section. Throughout this section, we assume that the domain <sup>Ω</sup> <sup>⊂</sup>*Rn*� is bounded and stellar relative to some sphere. Remember that a domain is called stellar relative to some point if any ray outgoing from this point has one and only one common point with the boundary of this domain. A domain is stellar with respect to some set if it is stellar at every point of this set. The following theorem is true.

**Theorem 1.25** If *qk*> *n*, then *W<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> compactly embedded in *<sup>C</sup>*ð Þ <sup>Ω</sup> *.*

The following theorem is true.

**Theorem 1.26** Let *qk*≤*n* and *Ω<sup>s</sup>* ⊂ *Ω*�be a piecewise smooth manifold of *s* dimensions, where *<sup>n</sup>* � *kq*<sup>&</sup>lt; *<sup>s</sup>*≤*n*. Then *<sup>W</sup><sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> continuously embedded in Lr*ð Þ *<sup>Ω</sup><sup>s</sup> (also in Lr*Þð Þ *<sup>Ω</sup><sup>s</sup>* ), where 1<sup>&</sup>lt; *<sup>r</sup>*<sup>&</sup>lt; *<sup>r</sup>*<sup>0</sup> <sup>¼</sup> *sq <sup>n</sup>*�*kq.*

The following theorems are proved in a completely similar way.

**Theorem 1.27** If *qk*≤*n*, then *W<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* compactly embedded in *Lr*ð Þ *<sup>Ω</sup>* (also in *Lr*Þð Þ *<sup>Ω</sup>* ), where 1≤ *r*< *nq <sup>n</sup>*�*kq* .

The following theorem is also true.

**Theorem 1.28** Let *u*∈*W<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>* . Then it has all possible generalized derivatives of any order *l*<*k* in *Ω*. At the same time *W<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> compactly embedded in C<sup>l</sup>* ð Þ *<sup>Ω</sup> , if* ð Þ *<sup>k</sup>* � *<sup>l</sup> <sup>q</sup>*>*<sup>n</sup>* and in *<sup>W</sup><sup>l</sup> <sup>r</sup>*ð Þ *<sup>Ω</sup> (also in W<sup>l</sup> <sup>r</sup>*Þð Þ *<sup>Ω</sup> ), if k*ð Þ � *<sup>l</sup> <sup>q</sup>*≤*<sup>n</sup>* and 1<sup>≤</sup> *<sup>r</sup>*<sup>&</sup>lt; *nq <sup>n</sup>*�*kq.*

Let us give some equivalent norms in the grand-Sobolev spaces *W<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> (then in *Nk <sup>q</sup>*Þð Þ <sup>Ω</sup> ). Let a function *<sup>f</sup>*ð Þ� continuous in *<sup>R</sup><sup>r</sup>* have the following properties

$$a) f(t) \ge 0 \land f(t) = \mathbf{0} \Leftrightarrow t = \mathbf{0};$$

$$(\beta) f(\lambda t) = |\lambda| f(t), \forall \lambda \in \mathbb{R} \land \forall t \in \mathbb{R}^r; \tag{59}$$

$$(\gamma) f(t + \tau) \le f(t) + f(\tau), \forall t; \tau \in \mathbb{R}^r.$$

In a completely analogous way to the classical case, the following theorem is proved.

**Theorem 1.29** *Let r denote the number of distinct monomials of degree* ≤ *k* � 1 *and let l*1, … , *lr be linear functionals bounded on W<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> that do not simultaneously vanish on any polynomial of degree* ≤*k* � 1*, except for the identically zero. Let f*ð Þ� *be a continuous function in R<sup>r</sup> , having the properties of a norm α*Þ � *γ*Þ*. Then the norm*

$$\left| \left| u \right| \right|\_{q}^{\*} \_{q} = f(l\_{1}u; l\_{2}u; \; \ldots; l\_{r}u) + \sum\_{|a|=k} \left| \left| \partial^{a}u \right| \right|\_{L\_{q}(\Omega)},\tag{60}$$

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

is equivalent to the norm k k� *<sup>W</sup><sup>k</sup> q*Þ ð Þ Ω .

The following lemma is true.

**Lemma 1.30** *Let Ω* ⊂ *Rn be a bounded domain with sufficiently smooth boundary ∂Ω. Then the norm* k k� <sup>∗</sup> *<sup>q</sup>*Þ,*k, defined by expression*

$$\left| \|u\|\right|\_{q),k}^{\*} = \sum\_{|a|=0}^{k-1} \left| \int\_{\partial\Omega} \partial^{a}u d\sigma \right| + \sum\_{|a|=k} \left||\partial^{a}u||\_{L\_{q}(\Omega)},\tag{61}$$

is equivalent to k k� *<sup>W</sup><sup>k</sup> q*Þ ð Þ <sup>Ω</sup> in *<sup>W</sup><sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> . In this case for *<sup>u</sup>*∈ ∘*W<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> the following inequality

$$||u||\_{L\_{q}(\Omega)} \leq \mathcal{C} \sum\_{|\alpha|=k} ||\partial^{\alpha}u||\_{L\_{q}(\Omega)},\tag{62}$$

holds, *C* is a constant independent of *u*.

#### **11. About the property (a). Fredholmness**

Let us get back to the question of whether property (A) is satisfied. Let Ω ⊂*R<sup>n</sup>* be a bounded domain. Let *<sup>S</sup>*<sup>⊂</sup> <sup>Ω</sup>� be some ð Þ *<sup>n</sup>* � <sup>1</sup> -dimensional surface. Define the following class of functions. Let *S*⊂ Ω� belong to class *C*ð Þ*<sup>k</sup>* and *ε*>0 be some number. Put Ω*ε*ð Þ¼ *S* f g *x*∈ Ω : *ρ*ð Þ *x*; *S* >*ε* .

We say that the function *f* belongs to class *C<sup>k</sup>* <sup>0</sup>ð Þ*S* (i.e. *f* vanishes in some neighborhood *S*), if *f* ∈*C<sup>k</sup>* ð Þ *<sup>Ω</sup>*� and <sup>∃</sup>*ε*<sup>&</sup>gt; <sup>0</sup> : *f x*ð Þ¼ 0, <sup>∀</sup>*x*<sup>∈</sup> *<sup>Ω</sup>*n*Ωε*ð Þ*<sup>S</sup> . Denote by* <sup>∘</sup>*N<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>*; *<sup>S</sup> the closure C<sup>k</sup>* <sup>0</sup>ð Þ*<sup>S</sup> in N<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup> .*

Thus, it is clear that ∘*N<sup>k</sup> <sup>q</sup>*Þð Þ¼ <sup>Ω</sup>; <sup>∂</sup><sup>Ω</sup> <sup>∘</sup>*N<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> . Denote by <sup>F</sup> *<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup>; *<sup>S</sup>* the factor space *N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> *<sup>=</sup>*∘*N<sup>m</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup>; *<sup>S</sup>* . Thus, <sup>F</sup> *<sup>m</sup> <sup>q</sup>*Þð Þ¼ <sup>Ω</sup>; <sup>∂</sup><sup>Ω</sup> <sup>F</sup> *<sup>m</sup> <sup>q</sup>*Þ. Each function *<sup>f</sup>* <sup>∈</sup> *<sup>N</sup>*<sup>1</sup> *<sup>q</sup>*Þð Þ <sup>Ω</sup> (also *<sup>f</sup>* <sup>∈</sup> *<sup>N</sup><sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup> ) has a (unique) trace *f =<sup>S</sup>* on *S*, and *f =<sup>S</sup>* ∈*Lq*Þð Þ*S* . Consider the following class of functions

$$N\_{q\backslash}^k(\mathbb{S}) = N\_{q\backslash}^k(\mathbb{\Omega})/\_{\mathbb{S}} = \left\{ f \in L\_q(\mathbb{S}) : \exists u \in N\_{q\backslash}^k(\mathbb{\Omega}) \Rightarrow f = u/\_{\mathbb{S}} \right\}.\tag{63}$$

The following theorem is true.

**Lemma 1.31** *Let S*<sup>⊂</sup> *<sup>Ω</sup>*�∧*S*∈*C*ð Þ*<sup>k</sup> be n*ð Þ � <sup>1</sup> *-dimensional surface. Then the linear spaces* F*<sup>k</sup> <sup>q</sup>*Þð Þ *<sup>Ω</sup>*; *<sup>S</sup>* and *<sup>N</sup><sup>k</sup> <sup>q</sup>*Þð Þ*<sup>S</sup> , k*≥1*, are isomorphic.*

It is not hard to see that if *f* ∈ ∘*N<sup>k</sup> <sup>q</sup>*Þð Þ <sup>Ω</sup>; *<sup>S</sup>* , then *<sup>f</sup> <sup>=</sup><sup>S</sup>* <sup>¼</sup> 0, <sup>∀</sup>*k*<sup>≥</sup> 1. Applying this lemma, completely similar to Theorem 1.19, we can prove the following.

**Theorem 1.32** *Let Ω* ⊂*Rn be a bounded domain with a boundary ∂Ω* ∈*C*ð Þ<sup>2</sup> *. Let L be a second-order elliptic operator with coefficients aij* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup>*� *, ai*, *<sup>a</sup>* <sup>∈</sup>*L*∞ð Þ *<sup>Ω</sup> ,* <sup>∀</sup>*i,j* <sup>¼</sup> 1,�*<sup>n</sup> defined by* expression (44)*. Let S*<sup>⊂</sup> *<sup>Ω</sup>*�∧*S*∈*C*ð Þ<sup>2</sup> *be some n*ð Þ � <sup>1</sup> *-dimensional surface*

*and Ω*<sup>0</sup> ⊂ ⊂ *Ω*⋃ *S (i.e. Ω*<sup>0</sup> ⊂ *Ω*⋃ *S) be an arbitrary domain. Then the following estimate holds true for* ∀*u* ∈ ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>*; *<sup>S</sup> :*

$$\|\|u\|\|\_{N\_{q}^{2}(\Omega')} \leq \mathcal{C} \left( \|Lu\|\_{N\_{q}(\Omega)} + \|u\|\_{N\_{q}(\Omega)} \right),\tag{64}$$

where the constant *C* depends only on the ellipticity constants of operator *L*, on the norms of the coefficients of *L* in *L*∞ð Þ Ω , on *S* and Ω<sup>0</sup> (is independent of *u*).

It is not hard to see that Theorem 1.19 is a particular case of this theorem, for this it is sufficient to take *S* ¼ ∂Ω. By Theorem 1.32 completely analogous to Theorem 9.14 of the monograph [31, p. 240] the following theorem is proved.

**Theorem 1.33** *Let Ω* ⊂*R<sup>n</sup> be a bounded domain with a boundary ∂Ω* ∈*C*ð Þ<sup>2</sup> *and L be an elliptic operator* (44) *with coefficients aij* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup> , ai*, *<sup>a</sup>* <sup>∈</sup> *<sup>L</sup>*∞ð Þ *<sup>Ω</sup> , i*, *<sup>j</sup>* <sup>¼</sup> 1,*n . Then the following estimate holds for* ∀*u* ∈ ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>* :

$$\|\|u\|\|\_{N\_{q\mid}(\Omega)} \le C \|Lu - \sigma u\|\|\_{N\_{q\mid}(\Omega)},\tag{65}$$

for ∀*σ* ≥ *σ*0, where *C*; *σ*<sup>0</sup> >0 are some constants that independent of *u*. The following theorem is the result of Theorems 1.20 and 1.33.

**Theorem 1.34** Let *<sup>L</sup>* be an elliptic operator (44) with coefficients *aij* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup> , ai*, *<sup>a</sup>*∈*L*∞ð Þ *<sup>Ω</sup>* , *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>n</sup>*. Let *<sup>Ω</sup>* <sup>⊂</sup> *Rn* be a bounded domain with a boundary *<sup>∂</sup><sup>Ω</sup>* <sup>∈</sup>*C*ð Þ<sup>2</sup> *, which has a property Δ<sup>q</sup>*<sup>Þ</sup> . Then <sup>∃</sup>*σ*<sup>0</sup> <sup>&</sup>gt; 0: the equation *Lu* � *<sup>σ</sup><sup>u</sup>* <sup>¼</sup> *<sup>f</sup>* is uniquely solvable for <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ *<sup>Ω</sup> in class* <sup>∘</sup>*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>* , <sup>∀</sup>*<sup>σ</sup>* <sup>≥</sup>*σ*0*.*

The Fredholm alternatives hold for the equation Lu ¼ f, i.e. the following main theorem is true.

**Theorem 1.35** Let *<sup>L</sup>* be an elliptic operator (44) with coefficients *aij* <sup>∈</sup>*C*ð Þ *<sup>Ω</sup> , ai*, *<sup>a</sup>*∈*L*∞ð Þ *<sup>Ω</sup>* , *<sup>i</sup>*, *<sup>j</sup>* <sup>¼</sup> 1, *<sup>n</sup>* and *<sup>Ω</sup>* <sup>⊂</sup>*R<sup>n</sup>* be a bounded domain with a boundary *<sup>∂</sup><sup>Ω</sup>* <sup>∈</sup>*C*ð Þ<sup>2</sup> *, which has a property Δ<sup>q</sup>*<sup>Þ</sup> . Then: i) if *KerL* <sup>¼</sup> 0 in <sup>∘</sup>*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>* , then the boundary value problem *Lu* <sup>¼</sup> *<sup>f</sup>*, *<sup>u</sup>=<sup>Γ</sup>* <sup>¼</sup> *<sup>φ</sup>*, has a unique solution for <sup>∀</sup>*φ*<sup>∈</sup> <sup>F</sup><sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup>* and <sup>∀</sup>*<sup>f</sup>* <sup>∈</sup> *Nq*Þð Þ *<sup>Ω</sup>* ; ii) *KerL* is a finite-dimensional subspace in ∘*N*<sup>2</sup> *<sup>q</sup>*Þð Þ *<sup>Ω</sup> .*

Regarding the proof of all these facts one can see the works [4, 6].

#### **12. On one spectral problem in Morrey-Smirnov space**

In this section we consider one spectral problem in Morrey-Smirnov space. Such spectral problems arise in the problem of vibrations of a loaded string with fixed ends is solved by applying the Fourier method (see [32–34]). Morrey space is also non separable space and we define its subspace in which the infinitely differentiable functions are dense. We prove that eigenfunctions of the considered spectral problem form a basis in this subspace after eliminating an arbitrary term from them.

We need some facts from the theory of Morrey-type spaces. Let Γ be some rectifiable Jordan curve on the complex plane *C:* By j j *M* <sup>Γ</sup> we denote the linear

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

Lebesgue measure of the set *M* ⊂Γ. By the Morrey- Lebesgue space *Lp*,*<sup>α</sup>*ð Þ <sup>Γ</sup> , 0<sup>≤</sup> *<sup>α</sup>*≤1, *<sup>p</sup>* <sup>≥</sup>1, we mean a normed space of all functions *<sup>f</sup>*ð Þ� measurable on <sup>Γ</sup> equipped with a finite norm k k*<sup>f</sup> <sup>L</sup>p*,*α*ð Þ <sup>Γ</sup> :

$$\|f\|\_{L^{p,q}(\Gamma)} = \sup\_{B} \left( |B\cap\Gamma|\_{\Gamma}^{a-1} \int\_{B\cap\Gamma} |f(\xi)|^{p} |d\xi| \right)^{\frac{1}{p}} < +\infty. \tag{66}$$

*Lp*,*<sup>α</sup>*ð Þ <sup>Γ</sup> is a Banach space and *Lp*,1ð Þ¼ <sup>Γ</sup> *Lp*ð Þ <sup>Γ</sup> , *<sup>L</sup>p*,0ð Þ¼ <sup>Γ</sup> *<sup>L</sup>*∞ð Þ <sup>Γ</sup> . The embedding *Lp*,*α*<sup>1</sup> ð Þ <sup>Γ</sup> <sup>⊂</sup>*Lp*,*α*<sup>2</sup> ð Þ <sup>Γ</sup> is valid for 0 <sup>≤</sup>*α*<sup>1</sup> <sup>≤</sup>*α*<sup>2</sup> <sup>≤</sup> 1 . Thus *Lp*,*<sup>α</sup>*ð Þ <sup>Γ</sup> <sup>⊂</sup> *Lp*ð Þ <sup>Γ</sup> , <sup>∀</sup>*α*<sup>∈</sup> ½ � 0, 1 , <sup>∀</sup>*p*<sup>≥</sup> 1. The case of <sup>Γ</sup> � ½ � *<sup>a</sup>*, *<sup>b</sup>* will be denoted by *Lp*,*<sup>α</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* .

Denote by *<sup>L</sup>*~*p*,*<sup>α</sup>* ð Þ *<sup>a</sup>*, *<sup>b</sup>* the linear subspace of *Lp*,*<sup>α</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* consisting of functions whose shifts are continuous in *Lp*,*<sup>α</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* , i.e. k k *<sup>f</sup>*ð Þ� � þ *<sup>δ</sup> <sup>f</sup>*ð Þ� *<sup>L</sup>p*,*α*ð Þ *<sup>a</sup>*,*<sup>b</sup>* ! 0 as *<sup>δ</sup>* ! <sup>0</sup>*:* The closure of *<sup>L</sup>*~*<sup>p</sup>*,*<sup>α</sup>* ð Þ *<sup>a</sup>*, *<sup>b</sup>* in *Lp*,*<sup>α</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* will be denoted by *<sup>M</sup><sup>p</sup>*,*<sup>α</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* . In [35] the following theorem is proved.

**Theorem 1.36** The exponential system *ei nt* � � *<sup>n</sup>*∈*<sup>Z</sup>* is the basis in *Mp*,*<sup>α</sup>*ð Þ �*π*, *<sup>π</sup>* , 1<*p* < þ ∞, 0 <*α* ≤1*:*

Using this theorem, it is easy to obtain the following

**Theorem 1.37** Each of the trigonometric systems f g *sinnx* <sup>∞</sup> *<sup>n</sup>*¼<sup>1</sup> and f g *cosnx* <sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup> forms the basis for *<sup>M</sup><sup>p</sup>*,*<sup>α</sup>*ð Þ 0, *<sup>π</sup>* , 1<sup>&</sup>lt; *<sup>p</sup>*<sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>, 0<sup>&</sup>lt; *<sup>α</sup>*≤1.

Consider a sample eigenvalue problem for the discontinuous second-order differential operator

$$-\boldsymbol{\eta}^{\prime\prime}(\boldsymbol{x}) = \lambda \boldsymbol{\eta}(\boldsymbol{x}), \quad \boldsymbol{x} \in \left(0, \frac{1}{3}\right) \cup \left(\frac{1}{3}, 1\right), \tag{67}$$

with the boundary conditions

$$\begin{aligned} y(\mathbf{0}) &= y(\mathbf{1}) = \mathbf{0}, \\ y\left(\frac{\mathbf{1}}{3} - \mathbf{0}\right) &= y\left(\frac{\mathbf{1}}{3} + \mathbf{0}\right), \\ y'\left(\frac{\mathbf{1}}{3} - \mathbf{0}\right) &- y'\left(\frac{\mathbf{1}}{3} + \mathbf{0}\right) = \lambda m y\left(\frac{\mathbf{1}}{3}\right), \end{aligned} \tag{68}$$

where *λ* is the spectral parameter, *m* is a non-zero complex number. Let us give some results from [36], which we will need throughout the paper.

**Lemma 1.38** [36] *The spectral problem* (67), (68) *has two series of eigenfunctions which are given by the following expressions*

$$y\_{1,n}(\boldsymbol{\kappa}) = \sin \mathfrak{Z}m\boldsymbol{\kappa}, \boldsymbol{\kappa} \in [0, 1], \quad n = 1, 2, \ldots,\tag{69}$$

$$\mathcal{Y}\_{2,n}(\mathbf{x}) = \begin{cases} \sin \rho\_{2,n} \left( \mathbf{x} - \frac{\mathbf{1}}{3} \right) + \sin \rho\_{2,n} \left( \mathbf{x} + \frac{\mathbf{1}}{3} \right), \mathbf{x} \in \left[ \mathbf{0}, \frac{\mathbf{1}}{3} \right], \\\ \sin \rho\_{2,n} (1 - \mathbf{x}), \mathbf{x} \in \left[ \frac{\mathbf{1}}{3}, \mathbf{1} \right], n = 0, 1, 2, \dots \end{cases} \tag{70}$$

Let us construct the operator *L*, which linearizes the problem (67), (68) in the direct sum *Lp*ð Þ 0, 1 <sup>⊕</sup>*C*. Denote by *<sup>W</sup>*<sup>2</sup> *<sup>p</sup>* 0, <sup>1</sup> 3 � �⊕*W*<sup>2</sup> *p* 1 <sup>3</sup> , 1 � � the space of functions whose restrictions to intervals 0, <sup>1</sup> 3 � � and <sup>1</sup> <sup>3</sup> , 1 � � belong to Sobolev spaces *<sup>W</sup>*<sup>2</sup> *<sup>p</sup>* 0, <sup>1</sup> 3 � � and *W*<sup>2</sup> *p* 1 <sup>3</sup> , 1 � �, respectively, where 1<*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* Let us define the operator *<sup>L</sup>* in the following way. As its domain *DL* we take the manifold

$$\begin{aligned} D\_{\mathbf{L}} &= \left\{ \hat{\boldsymbol{\mathcal{y}}} = \left( \boldsymbol{\mathcal{y}}(\mathbf{x}), \boldsymbol{m} \boldsymbol{\mathcal{y}} \left( \frac{\mathbf{1}}{3} \right) \right) : \boldsymbol{\mathcal{y}}(\mathbf{x}) \in \boldsymbol{W}\_{p}^{2} \left( \mathbf{0}, \frac{\mathbf{1}}{3} \right) \oplus \boldsymbol{W}\_{p}^{2} \left( \frac{\mathbf{1}}{3}, \mathbf{1} \right), \\ \boldsymbol{\mathcal{y}}(\mathbf{0}) = \boldsymbol{\mathcal{y}}(\mathbf{1}) &= \boldsymbol{0}, \boldsymbol{y} \left( \frac{\mathbf{1}}{3} - \mathbf{0} \right) = \boldsymbol{y} \left( \frac{\mathbf{1}}{3} + \mathbf{0} \right) \right\}, \end{aligned} \tag{71}$$

and for ^*y* ∈ *DL* the operator *L* is defined by the relation

$$L\hat{\jmath} = \left( -\jmath''; \jmath'\left(\frac{1}{3} - 0\right) - \jmath'\left(\frac{1}{3} + 0\right) \right). \tag{72}$$

The following lemma holds true.

**Lemma 1.39** The operator *L* defined by expressions (71), (72) is a densely defined closed operator with a completely continuous resolvent. The eigenvalues of the operator *L* and the problem (67), (68) coincide. If *y x*ð Þ *is the eigenfunction (associated function) of problem* (67), (68), then ^*<sup>y</sup>* <sup>¼</sup> *y x*ð Þ; *my* <sup>1</sup> 3 � � � � is the eigenvector (associated vector) of the operator *L:*

In order to obtain the main results, we need some concepts and facts from the theory of bases in a Banach space.

Recall the following definition.

**Definition 1.40** *The basis u*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup> of Banach space X is called a p-basis, if for any x*∈*X one has the inequality*

$$\left(\sum\_{n=1}^{\infty} |\langle \boldsymbol{\kappa}, \theta\_n \rangle|^p \right)^{\frac{1}{p}} \le M ||\boldsymbol{\kappa}||,\tag{73}$$

where f g *ϑ<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* is the biorthogonal system for f g *un <sup>n</sup>* <sup>∈</sup> *<sup>N</sup>*.

**Definition 1.41** *The sequences u*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup> and* f g *φ<sup>n</sup> <sup>n</sup>* <sup>∈</sup> *<sup>N</sup> of Banach space X are called pclose, if*

$$\sum\_{n=1}^{\infty} ||u\_n - \rho\_n||^p < \infty. \tag{74}$$

We will also use the following results from [37, 38] (see also [39, 40]).

**Theorem 1.42** [37] *Let x*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup> form a q-basis for the space X, and the system yn* � � *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup> is p- close to x*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *N, where* <sup>1</sup> *<sup>p</sup>* <sup>þ</sup> <sup>1</sup> *<sup>q</sup>* ¼ 1*. Then the following properties are equivalent:*


*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

3. *yn* � � *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* forms an isomorphic basis to f g *xn <sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* for *X*.

Let *<sup>X</sup>*<sup>1</sup> <sup>¼</sup> *<sup>X</sup>*⊕*C<sup>m</sup>* and f g *<sup>u</sup>*^*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* <sup>⊂</sup>*X*<sup>1</sup> be some minimal system, and *<sup>ϑ</sup>*^*<sup>n</sup>* � � *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* <sup>⊂</sup>*X*<sup>∗</sup> 1 ¼ *X*<sup>∗</sup> ⊕*Cm* be its biorthogonal system:

$$
\hat{u}\_n = (u\_n; a\_{n1}, \dots, a\_{nm}); \hat{\theta}\_n = (\theta\_n; \beta\_{n1}, \dots, \beta\_{nm}).\tag{75}
$$

Let *J* ¼ f g *n*1, … , *nm* be some set of *m* natural numbers. Suppose

$$\delta = \det \left\| \beta\_{n\bar{j}} \right\|\_{i,\bar{j}=\overline{1,m}}.\tag{76}$$

The following theorem holds true.

**Theorem 1.43** [38] *Let the system* f g *u*^*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *<sup>N</sup> form a basis for X*1*. In order to the system u*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *NJ , where NJ* ¼ *N*n*J, form a basis for X it is necessary and sufficient that the condition δ* 6¼ 0 *is satisfied. In this case the biorthogonal system to u*f g*<sup>n</sup> <sup>n</sup>*<sup>∈</sup> *NJ is defined by*

$$
\boldsymbol{\theta}\_{n}^{\*} = \frac{1}{\delta} \begin{vmatrix}
\boldsymbol{\theta}\_{n} & \boldsymbol{\theta}\_{n1} & \dots & \boldsymbol{\theta}\_{nm} \\
\boldsymbol{\beta}\_{n1} & \boldsymbol{\beta}\_{n1} & \dots & \boldsymbol{\beta}\_{n,1} \\
\dots & \dots & \dots & \dots \\
\boldsymbol{\beta}\_{nm} & \boldsymbol{\beta}\_{n1m} & \dots & \boldsymbol{\beta}\_{n,m}
\end{vmatrix} . \tag{77}$$

For *δ* ¼ 0 the system f g *un <sup>n</sup>*<sup>∈</sup> *NJ* is not complete and is not minimal in *X*. Let *<sup>X</sup>* be a Banach space and f g *ukn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* be some system in *<sup>X</sup>*. Let *a*ð Þ *<sup>n</sup> ik* , *i*, *k* ¼ 1, *m*, *n* ∈ *N*, be some complex numbers. Put

$$A\_n = \left(a\_{ik}^{(n)}\right)\_{i,k=\overline{1,n}} \text{ and } \Delta\_n = \det A\_n, n \in N. \tag{78}$$

Let us consider the following system in space *X*

$$
\hat{u}\_{kn} = \sum\_{i=1}^{m} a\_{ik}^{(n)} u\_{in}, \\
k = \overline{1, m}; n \in \mathcal{N}. \tag{79}
$$

**Theorem 1.44** *If the system u*f g *kn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* forms a basis for *<sup>X</sup>* and

$$
\Delta\_n \neq 0, \forall n \in \mathcal{N}, \tag{80}
$$

then the system f g *<sup>u</sup>*^*kn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* forms a basis with parentheses for *<sup>X</sup>*. If in addition the conditions

$$\sup\_{n} \{ \|A\_n\|, \left\|A\_n^{-1}\right\| \} < \infty, \sup\_{n} \{ \|u\_{kn}\|, \left\|\theta\_{kn}\right\| \} < \infty,\tag{81}$$

hold, where f g *<sup>ϑ</sup>kn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* <sup>⊂</sup>*X*<sup>∗</sup> is biorthogonal system to f g *ukn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>*, then the system f g *<sup>u</sup>*^*kn <sup>k</sup>*¼1,*m*; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* forms the usual basis for *<sup>X</sup>*.

The following theorem holds true.

**Theorem 1.45** The system of eigen and associated vectors of the operator *L* forms the basis for space *Mp*,*<sup>α</sup>*ð Þ 0, 1 <sup>⊕</sup>*C*, 1<*p*<sup>&</sup>lt; <sup>∞</sup>, 0 <sup>&</sup>lt;*<sup>α</sup>* <sup>≤</sup>1*:*

Now, let us consider the basicity of the system *y*<sup>0</sup> � �<sup>∪</sup> *yi*,*<sup>n</sup>* n o<sup>∞</sup> *i*¼1,2; *n*∈ *N* with a removed function in space *Mp*,*<sup>α</sup>*ð Þ 0, 1 *:*

**Theorem 1.46** *If from the system of eigen and associated functions of problem* (67), (68) *y*<sup>0</sup> � �<sup>∪</sup> *yi*,*<sup>n</sup>* n o<sup>∞</sup> *<sup>i</sup>*¼1,2; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup> we eliminate any function y*2,*n*<sup>0</sup> ð Þ *x , corresponding to a simple eigenvalue, then the new system forms a basis for Mp*,*<sup>α</sup>*ð Þ 0, 1 *,*1<*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, 0<*<sup>α</sup>* <sup>≤</sup>1*. And if we eliminate any function y*1,*n*<sup>0</sup> ð Þ *x from this system, then the obtained system does not form a basis in M<sup>p</sup>*,*<sup>α</sup>*ð Þ 0, 1 *; moreover, in this case this system is not complete and is not minimal in this space.*

*Proof.* For the eigenfunctions f g *<sup>z</sup>*<sup>0</sup> <sup>∪</sup>f g *zi*,*<sup>n</sup>* <sup>∞</sup> *<sup>i</sup>*¼1,2; *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* of the adjoint problem we have *z*1,*<sup>n</sup>* <sup>1</sup> 3 � � <sup>¼</sup> 0 for any *<sup>n</sup>* <sup>∈</sup> *<sup>N</sup>* and *<sup>z</sup>*2,*<sup>n</sup>* <sup>1</sup> 3 � � 6¼ 0. On the other hand, the eigenvectors of the adjoint operator *<sup>L</sup>*<sup>∗</sup> are defined by ^*zn* <sup>¼</sup> *zn*, *mzn* <sup>1</sup> 3 � � � � , *<sup>n</sup>* <sup>¼</sup> 0, 1, … , . Applying Theorem 1.43 to the system ^*y*<sup>0</sup> � �<sup>∪</sup> ^*yi*,*<sup>n</sup>* n o<sup>∞</sup> *i*¼1,2; *n*∈ *N* , we notice that *<sup>δ</sup>* <sup>¼</sup> *mz*1,*<sup>n</sup>* <sup>1</sup> 3 � � <sup>¼</sup> <sup>0</sup> for any *<sup>n</sup>*<sup>∈</sup> *<sup>N</sup>* and *<sup>δ</sup>* <sup>¼</sup> *mz*2,*<sup>n</sup>* <sup>1</sup> 3 � � 6¼ 0 for any eigenfunction corresponding to a simple eigenvalue, and the statements of the theorem follow from the corresponding statements of Theorem 1.43. Theorem is proved.

### **Conflict of interest**

The authors declare no conflict of interest.

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

#### **Author details**

Bilal Bilalov<sup>1</sup> \*†, Sabina Sadigova1,2† and Zaur Kasumov1†

1 Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan

2 Khazar University, Baku, Azerbaijan

\*Address all correspondence to: b\_bilalov@mail.ru

† These authors contributed equally.

© 2022 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**

[1] Cruz-Uribe DV, Fiorenza A. Variable Lebesgue spaces. Birkhauser, Springer; 2013

[2] Adams DR. Morrey Spaces. Switzherland: Springer; 2016

[3] Kokilashvili V, Meskhi A, Rafeiro H, Samko S. Integral operators in nonstandard function spaces. In: Variable Exponent Lebesgue and Amalgam Spaces. Vol. Volume 1. Springer; 2016

[4] Kokilashvili V, Meskhi A, Rafeiro H, Samko S. Integral operators in nonstandard function spaces. In: Variable Exponent Holder, Morrey–Campanato and Grand Spaces. Vol. Volume 2. Springer; 2016

[5] Harjulehto P, Hasto P. Orlicz Spaces Generalized Orlicz Spaces, 169 p. Springer; 2019

[6] Castillo RE, Rafeiro H. An Introductory Course in Lebesgue Spaces. Springer; 2016

[7] Bilalov BT, Gasymov TB, Guliyeva AA. On solvability of Riemann boundary value problem in Morrey-hardy classes. Turk. J. of Math. 2016;**40**(50):1085-1101. DOI: 10.3906/mat-1507-10

[8] Bilalov BT, Guseynov ZG. Basicity of a system of exponents with a piece-wise linear phase in variable spaces. Mediterranean Journal of Mathematics. 2012;**9**(3):487-498. DOI: 10.1007/ s00009-011-0135-7

[9] Bilalov BT, Guliyeva AA. On basicity of the perturbed systems of exponents in Morrey-Lebesgue space. International Journal of Mathematics. 2014;**25** (1450054):1-10

[10] Bilalov BT. The basis property of a perturbed system of exponentials in

Morrey-type spaces. Sib. Math. Journ. 2019;**60**(2):323-350

[11] Israfilov DM, Tozman NP. Approximation in Morrey-Smirnov classes. Azerb. J. Math. 2011;**1**(1): 99-113

[12] Sharapudinov II. On direct and inverse theorems of approximation theory In variable Lebesgue and Sobolev spaces. Azerb. J. Math. 2014;**4**(1):55-72

[13] Bilalov BT, Huseynli AA, El-Shabrawy SR. Basis properties of trigonometric Systems in Weighted Morrey Spaces. Azerb. J. Math. 2019; **9**(2):200-226

[14] Bilalov BT, Seyidova FS. Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces. Turkish Journal of Mathematics. 2019; **43**:1850-1866. DOI: 10.3906/mat-1901-113

[15] Zeren Y, Ismailov MI, Karacam C. Korovkin-type theorems and their statistical versions in grand Lebesgue spaces. Turkish Journal of Mathematics. 2020;**44**:1027-1041. DOI: 10.3906/mat-2003-21

[16] Zeren Y, Ismailov M, Sirin F. On basicity of the system of eigenfunctions of one discontinuous spectral problem for second order differential equation for grand-Lebesgue space. Turkish Journal of Mathematics. 2020;**44**(5):1995-1612. DOI: 10.3906/mat-2003-20

[17] Bilalov BT, Sadigova SR. On solvability in the small of higher order elliptic equations in grand-Sobolev spaces. Complex Variables and Elliptic Equations. 2020;**66**:2117-2130. DOI: 10.1080/17476933.2020.1807965

*Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces DOI: http://dx.doi.org/10.5772/intechopen.104918*

[18] Bilalov BT, Sadigova SR. Interior Schauder-type estimates for higherorder elliptic operators in grand-Sobolev spaces. Sahand Communications in Mathematical Analysis. 2021;**18**:129-148

[19] Palagachev DK, Softova LG. Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's. Potential Analysis. 2004;**20**:237-263

[20] Chen Y. Regularity of the solution to the Dirichlet problem in Morrey space. J. Partial Differ. Eqs. 2002;**15**:37-46

[21] Fario GD, Ragusa MA. Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients. Journ. of Func. Anal. 1993; **112**:241-256. DOI: 10.1006/ jfan.1993.1032

[22] Softova LG. The Dirichlet problem for elliptic equations with VMO coefficients in generalized Morrey spaces. Operator Theory: Advances and Applications. 2013;**229**:371-386. DOI: 10.1007/978-3-0348-0516-2\_21

[23] Palagachev DK, Ragusa MA, Softova LG. Regular obligue derivative problem in Morrey spaces, Elec. Jour. of Diff. Eq. 2020;**2000**(39):1-17 https://digital.libra ry.txstate.edu/handle/10877/9133

[24] Byun SS, Palagachev DK, Softova LG. Survey on gradient estimates for nonlinear elliptic equations in various function spaces. St. Petersbg. Math. J. 2020;**31**(3):401-419 and Algebra Anal. 31, No. 3, 10-35 (2019)

[25] Caso L, D'Ambrosio R, Softova L. Generalized Morrey spaces over unbounded domains. Azerb. J. Math. 2020;**10**(1):193-208

[26] Di Fazio G, Palagachev DK, Ragusa MA. Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients. Journal of Functional Analysis. 1999;**166**(2):179-196. DOI: 10.1006/jfan.1999.3425

[27] Di Fazio G. On Dirichlet problem in Morrey spaces. Differential and Integral Equations. 1993;**6**(2):383-391

[28] Bers L, John F, Schechter M. Partial differential equations. Moscow: Mir; 1966 (in Russian)

[29] Mikhaylov VP. Partial differential equations. Moscow: Nauka; 1976 (in Russian)

[30] Bilalov BT, Sadigova SR. On the Fredholmness of the Dirichlet problem for a second-order elliptic equation in grand-Sobolev spaces. Ricerche di Matematica. 2021. DOI: 10.1007/s11587- 021-00599-9

[31] Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order. Berlin-Heidelberg-New York-Tokyo: Springer-Verlag; 1983

[32] Atkinson FV. Discrete and Continuous Boundary Problems. Moscow: Mir; 1968

[33] Collatz L. Eigenvalue Problems. Moscow: Fizmatgiz; 1968 504 p (in Russian)

[34] Tikhonov AN, Samarskii AA. Equations of Mathematical Physics. Dover, New York: Mosk. Gos. Univ., Moscow (1999); 2011

[35] Bilalov BT, Quliyeva AA. On basicity of exponential systems in Morrey type spaces. International Journal of Mathematics. 2014;**25**(6):1-10

[36] Gasymov TB, Maharramova GV. On completeness of eigenfunctions of the

spectral problem. Caspian Journal of Applied Mathematics, Ecology and Economics. 2015;**3**(2):66-76

[37] Bilalov BT. Bases of exponentials, sines, and cosines. Differentsial0 nye Uravneniya. 2003;**39**(5):619-623

[38] Gasymov TB. On necessary and sufficient conditions of basicity of some defective systems in Banach spaces. Transactions of National Academy of Sciences Azerbaijan Series of Physical Technical and Mathematical Sciences. 2006;**26**(1):65-70

[39] Bilalov BT. Some Problems of Approximation. Baku, Elm; 2016 380 p. (in Russian)

[40] Bilalov BT, Guseynov ZG. K-Bessel and K-Hilbert systems and K-bases. Doklady Mathematics. 2009;**80**(3): 826-828

Section 2
