A Study for Coupled Systems of Nonlinear Boundary Value Problem

*Noureddine Bouteraa and Habib Djourdem*

#### **Abstract**

This chapter deals with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions and for the system of two-point boundary value problem when we take the case of integer derivative. The existence results for the fist problem are obtained by using Leray-Shauder nonlinear alternative and Banach contraction principle and for the second problem, we derive explicit eigenvalue intervals of *λ* for the existence of at least one positive solution by using Krasnosel'skii fixed point theorem. An illustrative examples is presented at the end for each problem to illustrate the validity of our results.

**Keywords:** positive solution, uniqueness, Green's function, system of fractional differential equations, system of differential equations, existence, nonlocal boundary value problem, fixed point theorem

#### **1. Introduction**

In this chapter, we are interested in the existence of solutions for the nonlinear fractional boundary value problem (BVP)

$$\begin{cases} \ ^cD^a u(t) = f(t, u(t), v(t)), & t \in [0, 1], \ 2 < a \le 3, \\ ^cD^\beta v(t) = g(t, u(t), v(t)), & t \in [0, 1], \ 2 < \beta \le 3, \\ \lambda u(0) + \gamma u(1) = u(\eta), & \lambda v(0) + \gamma v(1) = v(\eta), \\ u(0) = \int\_0^\eta u(s)ds, & v(0) = \int\_0^\eta v(s)ds, \\ \lambda^c D^\mu u(0) + \gamma^c D^\mu u(1) = ^cD^\mu u(\eta), \quad 1 < p \le 2. \end{cases} \tag{1}$$

We also study the integer case of problem

$$\begin{cases} \boldsymbol{u}^{(4)}(t) = \lambda \boldsymbol{a}(t)\boldsymbol{f}(\boldsymbol{v}(t)), & 0 < t < 1, \\ \boldsymbol{v}^{(4)}(t) = \lambda \boldsymbol{b}(t)\boldsymbol{g}(\boldsymbol{u}(t)), & 0 < t < 1, \\ \boldsymbol{u}(0) = \mathbf{0}, \boldsymbol{u}'(0) = \mathbf{0}, \boldsymbol{u}''(1) = \mathbf{0}, & \boldsymbol{u}'''(1) = \mathbf{0}, \\ \boldsymbol{v}(0) = \mathbf{0}, \boldsymbol{v}'(0) = \mathbf{0}, \boldsymbol{v}''(1) = \mathbf{0}, & \boldsymbol{v}'''(1) = \mathbf{0}. \end{cases} \tag{2}$$

where *D<sup>α</sup>* <sup>0</sup><sup>þ</sup> , *<sup>D</sup><sup>β</sup>* <sup>0</sup><sup>þ</sup> are the standard Riemann-Liouville fractional derivative of order *α* and *<sup>β</sup>*, the functions *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*<sup>C</sup>* ð Þ� 0, 1 <sup>2</sup> , � �, the functions *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*C*ð Þ ð Þ� 0, 1 , in the second problem and *λ*> 0, *a*, *b*∈*C*ð Þ ½ � 0, 1 , 0, ½ Þ ∞ .

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 [1–6] and the references cited therein.

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 modelled 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 [14–37] 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 [38, 39]. Let us cited few papers. Zhang et al. [40] studied the existence of two positive solutions of following singular fractional boundary value problems:

$$\begin{cases} D\_{0+}^{a}u(t) + f(t, u(t)) = \mathbf{0}, \quad t \in (\mathbf{0}, 1) \\\\ u(\mathbf{0}) = \mathbf{0}, \ D\_{0+}^{\beta}u(\mathbf{0}) = \mathbf{0}, \quad D\_{0+}^{\beta}u(\mathbf{1}) = \sum\_{j=1}^{\infty} D\_{0+}^{\beta}u\left(\eta\_{j}\right), \end{cases} \tag{3}$$

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

where *D<sup>α</sup>* <sup>0</sup>þ, *<sup>D</sup><sup>β</sup>* <sup>0</sup><sup>þ</sup> are the stantard Riemann-Liouville fractional derivative of order *α*∈ ð � 2, 3 , *β* ∈½ � 1, 2 , *f* ∈*C*ð Þ ½ �� 0, 1 , and *aj*, *η<sup>j</sup>* ∈ð Þ 0, 1 , *α* � *β* ≥1 with P*i*¼<sup>0</sup> <sup>∞</sup> *ajη α*�*β*�1 *<sup>j</sup>* <1.

In [41], the authors studied the boundary value problems of the fractional order differential equation:

$$\begin{cases} D\_{0+}^{a}u(t) = f(t, u(t)) = 0, \quad t \in (0, 1), \\ u(0) = 0, \ D\_{0+}^{\beta}u(1) = aD\_{0+}^{\beta}u(\eta), \end{cases} \tag{4}$$

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 stantard 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 2015, Alsulami et al. [42] studied the existence of solutions of the following nonlinear third-order ordinary differential inclusion with multi-strip boundary conditions

$$\begin{cases} \boldsymbol{u}^{(3)}(t) \in F(t, \boldsymbol{u}(t)), \ t \in (0, 1), \\\\ \boldsymbol{u}(0) = \mathbf{0}, \ \boldsymbol{u}'(0) = \mathbf{0}, \\\\ \boldsymbol{u}(1) = \sum\_{i=1}^{n-2} a\_{i\zeta\_i}^{\eta\_i} \boldsymbol{u}(s) ds, \\\\ \quad 0 < \zeta\_i < \eta\_i < 1, i = 1, 2, \dots, n-2, n \ge 3. \end{cases} \tag{5}$$

In 2017, Resapour et al. [43] investigated a Caputo fractional inclusion with integral boundary condition for the following problem

$$\begin{cases} \,^cD^\alpha u(t) \in F(t, u(t), ^cD^\beta u(t), u'(t)), \\\\ u(0) + u'(0) + ^cD^\beta u(0) = \int\_0^\eta u(s)ds, \\\\ u(1) + u'(1) + ^cD^\beta u(1) = \int\_0^\nu u(s)ds, \end{cases} \tag{6}$$

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*.

Inspired and motivated by the works mentioned above, The goal of this chapter is to establish the existence and uniqueness results for the nonlocal boundary value problem system (1) by using some well-known tools of fixed point theory such as Banach contraction principle and Leray-Shauder nonlinear alternative and the existence of at least one positive solution for the system of two-point boundary value problem (2) by using Krasnosel'skii fixed point theorem. The aim of the last results is to establish some simple criteria for the existence of single positive solutions of the BVPs (2) in explicit intervals for *λ*. The chapter is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel, for more details; see [44] and we give main results of problem (1). Finally, we give an example to illustrate our result. In Section 3, deals with main results of problem (2) and we give an example to illustrate our results.

#### **2. Existence and uniqueness results for problem (1)**

#### **2.1 Preliminaries**

In this section, we introduce some definitions and lemmas, see [2, 4, 44–46]. Definition 2.1. Let *α* >0, *n* � 1< *α*< *n*, *n* ¼ ½ �þ *α* 1 and *u* ∈*C*ð Þ ½ Þ 0, ∞ , . The Caputo derivative of fractional order *α* for the function *u* is defined by

$$\, ^cD^a u(t) = \frac{1}{\Gamma(n-a)} \frac{t}{t} \left(t - s\right)^{n-a-1} u^{(n)}(s) ds,\tag{7}$$

where Γð Þ� is the Eleur Gamma function.

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

$$I^a u(t) = \frac{1}{\Gamma(a)\_0}^t (t - s)^{a-1} u(s) ds, \quad t > 0,\tag{8}$$

where Γð Þ� is the Eleur Gamma function, provided that the right side is pointwise defined on 0, ð Þ ∞ .Lemma 2.1. *Let α*> 0, *n* � 1<*α* <*n and the function g* : ½ �! 0, *T be continuous for each T* >0*. Then, the general solution of the fractional differential equation c <sup>D</sup>αg t*ðÞ¼ <sup>0</sup> *is given by*

$$\mathbf{g}(t) = c\_0 + c\_1 t + \dots + c\_{n-1} t^{n-1},\tag{9}$$

where *c*0,*c*1, … ,*cn*�<sup>1</sup> are real constants and *n* ¼ ½ �þ *α* 1*.*

Also, in [19], authors have been proved that for each *T* >0 and *u* ∈*C*ð Þ ½ � 0, *T* we have

$$I^{a\varepsilon}D^{a}u(t) = u(t) + c\_0 + c\_1t + \dots + c\_{n-1}t^{n-1},\tag{10}$$

where *c*0,*c*1, … ,*cn*�<sup>1</sup> are real constants and *n* ¼ ½ �þ *α* 1.

#### **2.2 Existence results**

Let *X* ¼ f g *u t*ð Þ : *u t*ð Þ∈*C*ð Þ ½ � 0, 1 , endowed with the norm k k*u* ¼ sup *t*∈½ � 0, 1 j j *u t*ð Þ such

that k k*u* < ∞. Then ð Þ *X*, k k*:* is a Banach space and the product space ð Þ *X* � *X*, k k ð Þ *u*, *v* is also a Banach space equipped with the norm k k ð Þ *u*, *v* ¼ k k*u* þ k k*v* .

Throughout the first section, we let

$$\begin{aligned} M &= \frac{\Gamma(3-p)}{|\boldsymbol{\gamma}-\boldsymbol{\eta}^{2-p}|} \neq \mathbf{0}, |\boldsymbol{\lambda}+\boldsymbol{\gamma}-\mathbf{1}| \neq \mathbf{0}, |\boldsymbol{\gamma}-\boldsymbol{\eta}^{2}| \neq \mathbf{0}, \quad Q = \left| 2(\mathbf{1}-\boldsymbol{\eta})(\boldsymbol{\gamma}-\boldsymbol{\eta}) + \boldsymbol{\eta}^{2}|\boldsymbol{\lambda}+\boldsymbol{\gamma}-\mathbf{1}| \right| \neq \mathbf{0}, \\\ A(\mathbf{t}) &= |\boldsymbol{\Lambda}\_{1}(\mathbf{t})| = |\boldsymbol{\lambda}+\boldsymbol{\gamma}-\mathbf{1}| (\boldsymbol{\eta}^{2}+2(\mathbf{1}-\boldsymbol{\eta})\mathbf{t}), \\\ B(\mathbf{t}) &= |\boldsymbol{\Lambda}\_{2}(\mathbf{t})| = \left(\boldsymbol{\eta}^{3}|\boldsymbol{\lambda}+\boldsymbol{\gamma}-\mathbf{1}| + \mathbf{3}|\boldsymbol{\gamma}-\boldsymbol{\eta}^{2}|(\mathbf{1}-\boldsymbol{\eta})\right) \left(\boldsymbol{\eta}^{2}+2(\mathbf{1}-\boldsymbol{\eta})\mathbf{t}\right) - Q\left(\boldsymbol{\eta}^{3}+3(\mathbf{1}-\boldsymbol{\eta})\mathbf{t}^{2}\right), \end{aligned} \tag{11}$$

and

$$Q = \mathfrak{I}(\mathfrak{1} - \eta)(\mathfrak{y} - \eta) + \eta^2(\mathfrak{lambda} + \mathfrak{y} - \mathfrak{1}) \neq \mathbf{0}.\tag{12}$$

Lemma 2.2. *Let y*∈*C*ð Þ ½ � 0, 1 , *. Then the solution of the linear differential system*

$$\begin{cases} \,^c \boldsymbol{D}^\mu u(t) = \boldsymbol{y}(t), \, \, ^c \boldsymbol{D}^\beta v(t) = h(t), \, \, t \in [0, 1], \,\, 2 < \alpha, \beta \le 3 \\\\ \lambda u(0) + \gamma u(1) = v(\eta), \, \, \lambda v(0) + \gamma v(1) = u(\eta), \\\\ u(0) = \int\_0^\eta v(s) ds, \quad v(0) = \int\_0^\eta u(s) ds, \\\\ \,\, \lambda^\prime D^p u(0) + \gamma^\prime D^p u(1) = \, ^c D^p v(\eta), \quad 1 < p \le 2, \\\\ \,\, \lambda^\prime D^p v(0) + \gamma^\prime D^p v(1) = \, ^c D^p u(\eta), \quad 1 < p \le 2, \end{cases} \tag{13}$$

is equivalent to the system of integral equations

$$\begin{split} u(t) &= \int\_{0}^{t} \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} \mathscr{y}(s) ds + \frac{\mathbf{1}}{\mathbf{1} - \eta\_{0}} \left( \int\_{0}^{t} \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} h(\tau) d\tau \right) ds \\ &- \frac{\Lambda\_{1}(t)}{Q(1-\eta)} \int\_{0}^{t} \left( \int\_{0}^{s} \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} h(\tau) d\tau \right) ds \\ &- \frac{\Lambda\_{2}(t)M}{\Phi(1-\eta)Q} \left[ \int\_{0}^{\eta} \frac{(\eta-s)^{\beta-p-1}}{\Gamma(\beta-p)} h(s) ds - \gamma \int\_{0}^{1} \frac{(1-s)^{\alpha-p-1}}{\Gamma(\alpha-p)} \mathscr{y}(s) ds \right] \\ &+ \frac{\Lambda\_{1}(t)}{Q(\lambda+\gamma-1)} \left[ \int\_{0}^{\eta} \frac{(\eta-s)^{\beta-1}}{\Gamma(\beta)} h(s) ds - \gamma \int\_{0}^{1} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)} \mathscr{y}(s) ds \right], \end{split} \tag{14}$$

and

$$\begin{split} v(t) &= \int\_{0}^{t} \frac{(t-s)^{\beta-1}}{\Gamma(\beta)} y(s) ds + \frac{1}{1-\eta} \int\_{0}^{\eta} \left( \int\_{0}^{s} \frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)} h(\tau) d\tau \right) ds \\ &- \frac{\Lambda\_{1}(t)}{Q(1-\eta)} \int\_{0}^{\eta} \left( \int\_{0}^{s} \frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)} h(\tau) d\tau \right) ds \\ &- \frac{\Lambda\_{2}(t)M}{\Phi(1-\eta)Q} \left[ \int\_{0}^{\eta} \frac{(\eta-s)^{\alpha-p-1}}{\Gamma(\alpha-p)} h(s) ds - \gamma \int\_{0}^{1} \frac{(1-s)^{\beta-p-1}}{\Gamma(\alpha-p)} y(s) ds \right] \\ &+ \frac{\Lambda\_{1}(t)}{Q(\lambda+\gamma-1)} \left[ \int\_{0}^{\eta} \frac{(\eta-s)^{\alpha-1}}{\Gamma(\alpha)} h(s) ds - \gamma \int\_{0}^{1} \frac{(1-s)^{\beta-1}}{\Gamma(\alpha)} y(s) ds \right], \end{split} \tag{15}$$

where

$$
\Lambda\_1(t) = (\lambda + \gamma - \mathbf{1}) \left( \eta^2 + \mathbf{2} (\mathbf{1} - \eta) t \right), \tag{16}
$$

and

$$
\Lambda\_2(\mathfrak{t}) = \left(\eta^3(\mathbb{A} + \mathfrak{y} - \mathbf{1}) + \mathfrak{z}(\mathfrak{y} - \mathfrak{y}^2)(\mathfrak{1} - \mathfrak{y})\right)\left(\eta^2 + \mathfrak{z}(\mathfrak{1} - \mathfrak{y})\mathfrak{t}\right) - \mathbb{Q}\left(\eta^3 + \mathfrak{z}(\mathfrak{1} - \mathfrak{y})\mathfrak{t}^2\right).
$$

Proof. It is well known that the solution of equation *<sup>c</sup> <sup>D</sup><sup>α</sup>u t*ðÞ¼ *y t*ð Þ can be written as

$$u(t) = I^a y(t) + c\_0 + c\_1 t + c\_2 t^2,\tag{17}$$

$$v(t) = I^{\beta}h(t) + d\_0 + d\_1t + d\_2t^2,\tag{18}$$

where *c*0,*c*1,*c*<sup>2</sup> ∈ and and *d*0, *d*1,*d*<sup>2</sup> ∈ are arbitrary constants. Then, from (68) we have

$$u'(t) = I^{a-1}\mathbf{y}(t) + c\_1 + 2c\_2t,\tag{19}$$

and

$$\, ^cD^p u(t) = I^{a-p} \chi(t) + c\_2 \frac{2t^{2-p}}{\Gamma(3-p)}, \quad 1 < p \le 2. \tag{20}$$

By using the three-point boundary conditions, we obtain.

$$\begin{split} c\_{2} &= \frac{M}{2} \left( l^{\beta - p} \eta(\eta) - \eta^{\alpha - p} \eta(1) \right), \\ c\_{0} &= -\frac{2\eta^{2} (\lambda + \gamma - 1)}{2(1 - \eta)Q} \Bigg[ \left( \int\_{0}^{\frac{\tau}{\rho}} \frac{(\varsigma - \tau)^{\beta - 1}}{\Gamma(\beta)} h(\tau) d\tau \right) ds + \frac{1}{1 - \eta} \Bigg[ \left( \int\_{0}^{\frac{\tau}{\rho}} \frac{(\varsigma - \tau)^{\beta - 1}}{\Gamma(\beta)} h(\tau) d\tau \right) ds \\ &- \frac{(\eta^{2} [\eta^{3} (\lambda + \gamma - 1) + 3(\gamma - \eta^{2})(1 - \eta)] - \eta^{3} Q \var{M}}{2(1 - \eta)Q} \Bigg[ \left[ \frac{(\eta - \varsigma)^{\beta - p - 1}}{\Gamma(\beta - p)} h(\varsigma) ds \right. \\ &\left. - \gamma \left[ \frac{(1 - \varsigma)^{\alpha - p - 1}}{\Gamma(\alpha - p)} \jmath(\varsigma) ds \right] + \frac{\eta^{2}}{Q} \left[ \left[ \frac{(\eta - \varsigma)^{\beta - 1}}{\Gamma(\beta)} h(\varsigma) ds - \gamma \right] \frac{(1 - \varsigma)^{\alpha - 1}}{\Gamma(\alpha)} \jmath(\varsigma) ds \right], \end{split} \tag{21}$$

and

$$\begin{split} c\_{1} &= \frac{-2(\lambda+\eta-1)}{Q} \Bigg[ \Bigg( \Bigg[ \frac{s}{\Gamma} \Bigg( \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} h(\tau) d\tau \bigg) ds \\ &- \frac{(\eta^{3}(\lambda+\eta-1)+3(\nu-\eta^{2})(1-\eta))M}{3Q} \Bigg[ \Bigg[ \Bigg{\frac{\eta}{\Gamma} \frac{(\eta-s)^{\beta-p-1}}{\Gamma(\beta-p)} h(s) ds - \gamma \Bigg{\frac{1}{\nu} \frac{(1-s)^{\alpha-p-1}}{\Gamma(\alpha-p)} \nu(s) ds \Bigg{]} \\ &+ \frac{2(1-\eta)}{Q} \Bigg{[ \Bigg{\frac{\eta}{\Gamma} \frac{(\eta-s)^{\beta-1}}{\Gamma(\beta)} \nu(s) ds - \gamma \Bigg{[} \frac{1}{\Gamma(\alpha)} \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)} \nu(s) ds \Bigg{]} .\end{split} \tag{22}$$

Substituting the values of constants *c*0,*c*<sup>1</sup> and *c*<sup>2</sup> in (68), we get solution (64). Similarly, we obtain solution (65). The proof is complete.

The following rolations hold:

$$|\mathbf{A}(t)| \le |\beta + \gamma - \mathbf{1}| \left(\eta^2 + \mathbf{2}(\mathbf{1} - \eta)\right) = A\_1,\tag{23}$$

and

$$|B(t)| \le \left| \left( \eta^3 |\beta + \eta - \mathbb{1}| + \mathfrak{Z} |\eta - \eta^2| (\mathbb{1} - \eta) \right) \left( \eta^2 + \mathfrak{Z} (\mathbb{1} - \eta) \right) - Q \left( \eta^3 + \mathfrak{Z} (\mathbb{1} - \eta) \right) \right| = B\_1,\tag{24}$$

For the sake of brevity, we set

$$\begin{split} \Delta\_{1} &= \frac{\eta^{\beta+1}}{(1-\eta)\Gamma(\beta+2)} + \frac{A\_{1}\eta^{\beta+1}}{Q(1-\eta)\Gamma(\beta+2)} + \frac{MB\_{1}\eta^{\beta-p}}{(1-\eta)Q\Gamma(\lambda-p+1)} \\ &+ \frac{A\_{1}\eta^{\beta}}{Q|\beta+\gamma-1|\Gamma(\beta+1)} \\ \Delta\_{2} &= \frac{MB\_{1}\eta}{6(1-\eta)Q\Gamma(a-p+1)} + \frac{A\_{1}\eta}{Q|\lambda+\gamma-1|\Gamma(a+1)} + \frac{1}{\Gamma(a+1)} \\ \Delta\_{3} &= \frac{\eta^{a+1}}{(1-\eta)\Gamma(a+2)} + \frac{A\_{1}\eta^{a+1}}{Q(1-\eta)\Gamma(a+2)} + \frac{MB\_{1}\eta^{a-p}}{(1-\eta)Q\Gamma(a-p+1)} \\ &+ \frac{A\_{1}\eta^{a}}{Q|\lambda+\gamma-1|\Gamma(a+1)}, \end{split} \tag{25}$$

and

$$\Delta\_{4} = \frac{MB\_{1}\gamma}{\Phi(1-\eta)Q\Gamma(\beta-p+1)} + \frac{A\_{1}\gamma}{Q|\lambda+\gamma-1|\Gamma(\beta+1)} + \frac{1}{\Gamma(\beta+1)}.\tag{26}$$

In view of Lemma 2, we define the operator *T* : *X* � *X* ! *X* � *X* by

$$T(\mu, \upsilon)(t) = \begin{pmatrix} T\_1(\mu, \upsilon)(t) \\ T\_2(\mu, \upsilon)(t) \end{pmatrix},\tag{27}$$

where

$$\begin{split} T\_{1}(u,v)(t) &= \left[ \frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)} f(s,u(s),v(s))ds + \frac{1}{1-\eta} \right] \left( \left[ \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} g(\tau,u(\tau),v(\tau))d\tau \right] ds \right. \\ &\left. - \frac{B(t)M}{\Theta(1-\eta)Q} \right] \left[ \frac{\left(\eta-s\right)^{\beta-p-1}}{\Gamma(\beta-p)} g(s,u(s),v(s))ds - \gamma \left[ \frac{(1-s)^{\alpha-p-1}}{\Gamma(\alpha-p)} f(s,u(s),v(s))ds \right] \right] \\ &+ \frac{A(t)}{Q(\beta+\gamma-1)} \left[ \left[ \frac{(\eta-s)^{\beta-1}}{\Gamma(\beta)} g(s,u(s),v(s))ds - \gamma \left[ \frac{(1-s)^{\alpha-1}}{\Gamma(\alpha)} f(s,u(s),v(s))ds \right. \right] \right. \\ &\left. - \frac{A(t)}{Q(1-\eta)} \right] \left[ \left( \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} g(\tau,u(\tau),v(\tau))d\tau \right)ds \right. \end{split} \tag{28}$$

and

$$\begin{split} T\_{2}(u,v)(t) &= \left[ \frac{(t-s)^{\beta-1}}{\Gamma(\beta)} g(s,u(s),v(s)) ds + \frac{1}{1-\eta} \right] \left( \left[ \frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)} f(\tau,u(\tau),v(\tau)) d\tau \right] ds \right. \\\\ &\left. - \frac{B(t)M}{\Theta(1-\eta)Q} \right] \left[ \frac{(\eta-s)^{\alpha-p-1}}{\Gamma(\alpha-p)} f(s,u(s),v(s)) ds - \gamma \left[ \frac{(1-s)^{\beta-p-1}}{\Gamma(\beta-p)} g(s,u(s),v(s)) ds \right] \right] \\\\ &\left. + \frac{A(t)}{Q(\beta+\gamma-1)} \right[ \left[ \frac{(\eta-s)^{\alpha-1}}{\Gamma(\alpha)} f(s,u(s),v(s)) ds - \gamma \left[ \frac{(1-s)^{\beta-1}}{\Gamma(\beta)} g(s,u(s),v(s)) ds \right. \end{split} \tag{29}$$

$$-\frac{A(t)}{Q(1-\eta)} \left[ \left( \frac{(s-\tau)^{\alpha-1}}{\Gamma(\alpha)} f(\tau,u(\tau),v(\tau)) d\tau \right) ds.$$

Observe that the boundary value problem (1) has solutions if the operator equation ð Þ¼ *u*, *v T u*ð Þ , *v* has fixed points.

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

Lemma 2.3. [44] *(Leray-Schauder alternative). Let E be a Banach space and T* : *E* ! *E be a completely continuous operator (i.e., a map restricted to any bounded set in E is compact). Let*

$$\varepsilon = \{ (u, v) \in X \times X : (u, v) = \lambda T(u, v), for some \, 0 < \lambda < 1\}. \tag{30}$$

Then either the *ε*ð Þ *T* is unbounded or *T* has at least one fixed point.

Theorem 1.1 Assume that *f*, *g* : ½ �� 0, 1 � ! are a continuous function and. ð Þ *H*<sup>1</sup> there exist a function *ki* ≥0*mi* ≥0, *i* ¼ 1, 2 and *k*<sup>0</sup> > 0, *m*<sup>0</sup> >0 such that ∀*u*∈ , ∀*v*∈ , *i* ¼ 1, 2, we have

$$|f(t, u, v)| \le k\_0 + k\_1|u| + k\_2|v|,\tag{31}$$

and

$$|\lg(t, u, v)| \le m\_0 + m\_1|u| + m\_2|v|.\tag{32}$$

If ð Þ *Δ*<sup>2</sup> þ *Δ*<sup>3</sup> *k*<sup>1</sup> þ ð Þ *Δ*<sup>1</sup> þ *Δ*<sup>4</sup> *m*<sup>1</sup> <1 and ð Þ *Δ*<sup>2</sup> þ *Δ*<sup>3</sup> *k*<sup>2</sup> þ ð Þ *Δ*<sup>1</sup> þ *Δ*<sup>4</sup> , *m*<sup>3</sup> <1, where *Δi*, *i* ¼ 1, 2, 3, 4 are given above. Then the boundary value problem (1)–(58) has at least one solution on [0,1].

Proof. It is clear that *T* is a continuous operator where *T* : *X* � *X* ! *X* � *X* is defined above. Now, we show that *T* is completely continuous. Let Ω ⊂*X* � *X* be bounded. Then there exist positive constants *L*<sup>1</sup> and *L*<sup>2</sup> such that

$$|f(t, u(t), v(t))| \le L\_1, \quad |\mathbf{g}(t, u(t), v(t))| \le L\_2, \quad \forall (u, v) \in \Omega. \tag{33}$$

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

Then for any ð Þ *u*, *v* ∈ Ω, we have

j j *T*1ð Þ *u*, *v* ð Þ*t* ≤ *L*2 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* <sup>þ</sup> j j *A t*ð Þ *<sup>L</sup>*<sup>2</sup> *Q*ð Þ 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* þ *L*<sup>1</sup> ð*t* 0 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* þ *MB t* j j ð Þ 6 1ð Þ � *<sup>η</sup> <sup>Q</sup> <sup>L</sup>*<sup>2</sup> ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�*p*�<sup>1</sup> Γð Þ *β* � *p ds* þ *γL*<sup>1</sup> ð 0 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�*p*�<sup>1</sup> Γð Þ *α* � *p ds* 2 4 3 5 <sup>þ</sup> j j *A t*ð Þ *Q*j j *λ* þ *γ* � 1 *L*2 ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> ds* <sup>þ</sup> *<sup>γ</sup>L*<sup>1</sup> ð1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 2 4 3 5, ≤*L*<sup>2</sup> 1 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* þ *A*1 *Q*ð Þ 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *αβ <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* 8 < : þ *MB*<sup>1</sup> 6 1ð Þ � *η Q* ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�*p*�<sup>1</sup> Γð Þ *β* � *p ds* þ *A*1 6j j *λ* þ *γ* � 1 ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> ds* 9 = ; þ*L*<sup>1</sup> *MγB*<sup>1</sup> 6 1ð Þ � *η Q* ð1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�*p*�<sup>1</sup> Γð Þ *α* � *p ds* þ *A*1*γ Q*j j *λ* þ *γ* � 1 ð1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 8 < : þ ð*t* 0 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 9 = ;, ≤*L*2*Δ*<sup>1</sup> þ *L*1Δ2*:* (34)

Hence

$$\|\|T\_1(u,v)\|\| \le L\_2 \Delta\_1 + L\_1 \Delta\_2. \tag{35}$$

In the same way, we can obtain that

$$\|\|T\_2(u,v)\|\| \le L\_1\Delta\_3 + L\_2\Delta\_4. \tag{36}$$

Thus, it follows from (78) and (95) that the operator *T* is uniformly bounded, since k k *T u*ð Þ , *v* ≤ *L*1ð Þþ *Δ*<sup>1</sup> þ *Δ*<sup>3</sup> *L*2ð Þ Δ<sup>2</sup> þ *Δ*<sup>4</sup> . Now, we show that *T* is equicontinuous. Let *t*1, *t*<sup>2</sup> ∈½ � 0, 1 with *t*<sup>1</sup> <*t*2. Then we have

$$\begin{split} \left| \left| T\_{1}(u(t\_{2}), v(t\_{2})) - T\_{1}(u(t\_{1}), v(t\_{1})) \right| \leq L\_{1} \right| \frac{\mathbb{I} \left( t\_{2} - s \right)^{a-1} - \left( t\_{1} - s \right)^{a-1}}{\Gamma(a)} ds \\ + L\_{1} \int\_{t\_{1}}^{t\_{2}} \frac{\left( t\_{2} - s \right)^{a-1}}{\Gamma(a)} ds + \frac{\left| A(t\_{2}) - A(t\_{1}) \right| L\_{2}}{Q(1 - \eta)} \int\_{0}^{s} \left( \frac{\left( s - \tau \right)^{\beta - 1}}{\Gamma(\beta)} d\tau \right) ds \\ + \frac{\left( B(t\_{2}) - B(t\_{1}) \right) M}{6(1 - \eta)Q} \left[ L\_{2} \int\_{0}^{\eta} \frac{\left( \eta - s \right)^{\beta - p - 1}}{\Gamma(\beta - p)} ds + \gamma L\_{1} \int\_{0}^{1} \frac{(1 - s)^{a - p - 1}}{\Gamma(a - p)} ds \right] \\ + \frac{A(t\_{2}) - A(t\_{1})}{Q|\lambda + \gamma - 1|} \left[ L\_{2} \int\_{0}^{\eta} \frac{(\eta - s)^{\beta - 1}}{\Gamma(\beta)} ds - \gamma L\_{1} \right] \frac{\left( 1 - s \right)^{a - 1}}{\Gamma(a)} ds \right]. \end{split} \tag{37}$$

Obviously, the right-hand side of the above inequality tends to zero as *t*<sup>2</sup> ! *t*1. Similarly, we have

$$\begin{split} \left| T\_{2}(u(t\_{2}),v(t\_{2})) - T\_{2}(u(t\_{1}),v(t\_{1})) \right| &\leq L\_{2} \left[ \frac{(t\_{2}-s)^{\beta-1} - (t\_{1}-s)^{\beta-1}}{\Gamma(\beta)} ds \right. \\ &\left. + L\_{2} \int\_{t\_{1}}^{t\_{2}} \frac{(t\_{2}-s)^{\beta-1}}{\Gamma(\beta)} ds + \frac{|A(t\_{2}) - A(t\_{1})|L\_{1}}{Q(1-\eta)} \right] \left( \left[ \frac{(s-\tau)^{a-1}}{\Gamma(a)} ds \right] ds \right. \\ &\left. + \frac{(B(t\_{2}) - B(t\_{1}))M}{6(1-\eta)Q} \right] L\_{1} \left[ \left. \frac{\eta}{\Gamma} \right| \frac{(\eta-s)^{a-p-1}}{\Gamma(a-p)} ds + \gamma L\_{2} \left[ \frac{(1-s)^{\beta-p-1}}{\Gamma(\beta-p)} ds \right] \right. \\ &\left. + \frac{A(t\_{2}) - A(t\_{1})}{Q|\lambda+\gamma-1|} \left[ L\_{1} \left[ \frac{\eta}{\Gamma(a)} \frac{\eta-s}{\Gamma(a)} ds - \gamma L\_{2} \right] \frac{1}{\Gamma(\beta)} \frac{(1-s)^{\beta-1}}{\Gamma(\beta)} ds \right]. \end{split} \tag{38} \right] (38) \end{split}$$

Again, it is seen that the right-hand side of the above inequality tends to zero as *t*<sup>2</sup> ! *t*1. Thus, the operator *T* is equicontinuous.

Therefore, the operator *T* is completely continuous.

Finally, it will be verified that the set *ε* ¼ f g ð Þ *u*, *v* ∈*X* � *X* : ð Þ¼ *u*, *v λT u*ð Þ , *v* , 0≤*λ*≤ 1 is bounded. Let ð Þ *u*, *v* ∈*ε*, with ð Þ¼ *u*, *v λT u*ð Þ , *v* for any *t*∈½ � 0, 1 , we have

$$u(t) = \lambda T\_1(u, v)(t), \quad v(t) = \lambda T\_2(u, v)(t). \tag{39}$$

Then

$$\begin{split} |u(t)| \leq &\Delta\_2(k\_0 + k\_1|u| + k\_2|v|) + \Delta\_1(m\_0 + m\_1|u| + m\_2|v|), \\ = &\Delta\_2 k\_0 + \Delta\_1 m\_0 + (\Delta\_2 k\_1 + \Delta\_1 m\_1)|u| + (\Delta\_2 k\_2 + \Delta\_1 m\_2)|v|, \end{split} \tag{40}$$

and

$$\begin{split} |\boldsymbol{v}(t)| &\leq \Delta\_{3} (k\_{0} + k\_{1}|\boldsymbol{u}| + k\_{2}|\boldsymbol{v}|) + \Delta\_{4} (m\_{0} + m\_{1}|\boldsymbol{u}| + m\_{2}|\boldsymbol{v}|), \\ &= \Delta\_{3} k\_{0} + \Delta\_{4} m\_{0} + (\Delta\_{3} k\_{1} + \Delta\_{4} m\_{1})|\boldsymbol{u}| + (\Delta\_{3} k\_{2} + \Delta\_{4} m\_{2})|\boldsymbol{v}|. \end{split} \tag{41}$$

Hence we have

$$||\boldsymbol{\mu}|| = \Delta\_2 k\_0 + \Delta\_1 m\_0 + (\Delta\_2 k\_1 + \Delta\_1 m\_1)||\boldsymbol{\mu}|| + (\Delta\_2 k\_2 + \Delta\_1 m\_2)||\boldsymbol{\nu}||,\tag{42}$$

and

$$\Delta||\nu|| = \Delta\_3 k\_0 + \Delta\_4 m\_0 + (\Delta\_3 k\_1 + \Delta\_4 m\_1)|u| + (\Delta\_3 k\_2 + \Delta\_4 m\_2)|v|,\tag{43}$$

which imply that

$$\begin{split} \|\boldsymbol{u}\|\| + \|\boldsymbol{v}\|\| &= (\Delta\_2 + \Delta\_3)k\_0 + (\Delta\_1 + \Delta\_4)m\_0 + [(\Delta\_2 + \Delta\_3)k\_1 + (\Delta\_1 + \Delta\_4)m\_1] \|\boldsymbol{u}\| \\ &+ [(\Delta\_2 + \Delta\_3)k\_2 + (\Delta\_1 + \Delta\_4)m\_2] \|\boldsymbol{v}\| .\end{split} \tag{44}$$

Consequently,

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

$$\left\|(\mathfrak{u},\mathfrak{v})\right\| = \frac{(\mathfrak{A}\_2 + \mathfrak{A}\_3)k\_0 + (\mathfrak{A}\_1 + \mathfrak{A}\_4)m\_0}{\mathfrak{A}\_0},\tag{45}$$

where

$$\Delta\_0 = \min \left\{ \mathbf{1} - [(\Delta\_2 + \Delta\_3)\mathbf{k}\_1 + (\Delta\_1 + \Delta\_4)\mathbf{m}\_1], \mathbf{1} - [(\Delta\_2 + \Delta\_3)\mathbf{k}\_2 + (\Delta\_1 + \Delta\_4)\mathbf{m}\_2] \right\}, \tag{46}$$

which proves that *ε* is bounded. Thus, by Lemma 15, the operator *T* has at least one fixed point. Hence boundary value problem (1) has at least one solution. The proof is complete.

Now, we are in a position to present the second main results of this paper.

Theorem 1.2 Assume that *<sup>f</sup>*, *<sup>g</sup>* : ½ �� 0, 1 <sup>2</sup> ! are continuous functions and there exist positive constants *L*<sup>1</sup> and *L*<sup>2</sup> such that for all *t*∈½ � 0, 1 and *ui*, *vi* ∈ , *i* ¼ 1, 2, we have.

$$\begin{aligned} \mathbf{1}. \left| f(t, u\_1, u\_2) - f(t, v\_1, v\_2) \right| &\le L\_1(|u\_1 - v\_1| + |u\_2 - v\_2|), \\\\ \mathbf{2}. \left| g(t, u\_1, u\_2) - g(t, v\_1, v\_2) \right| &\le L\_2(|u\_1 - v\_1| + |u\_2 - v\_2|). \end{aligned}$$

Then the boundary value problem (1) has a unique solution on [0,1] provided

$$(\Delta\_1 + \Delta\_3)L\_1 + (\Delta\_2 + \Delta\_4)L\_2 < \mathbf{1}.\tag{47}$$

Proof. Let us set *sup t*∈½ � 0, 1 j j *f t*ð Þ , 0, 0 ¼ *N*<sup>1</sup> < ∞ and *sup t*∈½ � 0, 1 j j *g t*ð Þ , 0, 0 ¼ *N*<sup>2</sup> < ∞. For *u* ∈*X*, we observe that

$$|f(t, u(t), v(t))| \le |f(t, u(t)) - f(t, 0, 0)| + |f(t, 0, 0)|,$$

$$\le L\_1(|u(t)| + |v(t)|) + N\_1,\tag{48}$$

$$\le L\_1(||u|| + ||v||) + N\_1,$$

and

$$|\lg(t, u(t), v(t))| \le |\lg(t, u(t)) - \lg(t, \mathbf{0}, \mathbf{0})| + |\lg(t, \mathbf{0}, \mathbf{0})| \le L\_2 \|u\| + N\_2. \tag{49}$$

Then for *u* ∈*X*, we have

$$\begin{split} &|T\_{1}(\boldsymbol{u},\boldsymbol{v})(t)| \leq \frac{1}{1-\eta} \Big[ \Big( \int\_{0}^{t} \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} [L\_{2}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{2}] d\tau \Big) ds \\ &+ \frac{|A(t)|}{Q(1-\eta)} \Big[ \int\_{0}^{t} \Big( \frac{(s-\tau)^{\beta-1}}{\Gamma(\beta)} [L\_{2}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{2}] d\tau \Big) ds + \int\_{0}^{t} \frac{(t-s)^{a-1}}{\Gamma(a)} [L\_{1}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{1}] ds \\ &+ \frac{M|B(t)|}{6(1-\eta)Q} \Bigg[ \int\_{0}^{\eta} \frac{(\eta-s)^{\beta-p-1}}{\Gamma(\beta-p)} [L\_{2}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{2}] ds + \gamma \Big[ \frac{(1-s)^{a-p-1}}{\Gamma(a-p)} [L\_{1}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{1}] ds \Big] \\ &+ \frac{|A(t)|}{Q|\boldsymbol{\lambda}+\boldsymbol{\gamma}-\boldsymbol{1}|} \Bigg[ \int\_{0}^{\frac{\eta}{\boldsymbol{\gamma}}} \frac{(\eta-s)^{\beta-1}}{\Gamma(\beta)} [L\_{2}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{2}] ds + \gamma \Big[ \frac{(1-s)^{a-1}}{\Gamma(a)} [L\_{1}](\boldsymbol{u},\boldsymbol{v}) ||+N\_{1}] ds, \end{split}$$

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

≤ð Þ *L*2k k ð Þ *u*, *v* þ *N*<sup>2</sup> 1 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> Γð Þ *β dτ* 0 @ 1 A*ds* þ *A*1 *Q*ð Þ 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> Γð Þ *β dτ* 0 @ 1 A*ds* 8 < : þ *MB*<sup>1</sup> 6 1ð Þ � *η Q* ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�*p*�<sup>1</sup> Γð Þ *β* � *p ds* þ *A*1 6j j *λ* þ *γ* � 1 ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> ds* 9 = ; þ *MB*<sup>1</sup> 6 1ð Þ � *η Q* ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�*p*�<sup>1</sup> Γð Þ *β* � *p ds* þ *A*1 6j j *λ* þ *γ* � 1 ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> ds* 9 = ; þð Þ *L*1k k ð Þ *u*, *v* þ *N*<sup>1</sup> ð*t* 0 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 8 < : þ *MγB*<sup>1</sup> 6 1ð Þ � *η Q* ð1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�*p*�<sup>1</sup> Γð Þ *α* � *p ds* þ *A*1*γ Q*j j *λ* þ *γ* � 1 ð1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 9 = ;, ≤ð Þ *L*2*r* þ *N*<sup>2</sup> *Δ*<sup>1</sup> þ ð Þ *L*1*r* þ *N*<sup>1</sup> Δ<sup>2</sup> (50)

Hence

$$\|\|T\_1(u,v)\|\| \le (L\_2\Delta\_1 + L\_1\Delta\_2)r + N\_2\Delta\_1 + N\_1\Delta\_2\tag{51}$$

In the same way, we can obtain that

$$\left\|\left|T\_{2}(u,v)\right\|\right\| \leq (L\_{1}\Delta\_{3} + L\_{2}\Delta\_{4})r + N\_{2}\Delta\_{4} + N\_{1}\Delta\_{3}.\tag{52}$$

Consequently,

$$\left| \left| T(\mathfrak{u}, \boldsymbol{\nu}) \right| \right| \leq ( (\Delta\_2 + \Delta\_3) L\_1 + (\Delta\_1 + \Delta\_4) L\_2 ) r + N\_2 (\Delta\_1 + \Delta\_4) + N\_1 (\Delta\_2 + \Delta\_3) \leq r. \tag{53}$$

Now, for ð Þ *u*1, *v*<sup>1</sup> ,ð Þ *u*2, *v*<sup>2</sup> ∈*X* � *X* and for each *t*∈½ � 0, 1 , it follows from assumption ð Þ *H*<sup>3</sup> that

j j *T*1ð Þ *u*2, *v*<sup>2</sup> ðÞ�*t T*1ð Þ *u*1, *v*<sup>1</sup> ð Þ*t* ≤ *L*2ð Þ k k *u*<sup>2</sup> � *u*<sup>1</sup> þ k k *v*<sup>2</sup> � *v*<sup>1</sup> 1 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* 8 < : þ *A*1 *Q*ð Þ 1 � *η* ð *η* 0 ð*s* 0 ð Þ *<sup>s</sup>* � *<sup>τ</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> <sup>d</sup><sup>τ</sup>* 0 @ 1 A*ds* þ *MB*<sup>1</sup> 6 1ð Þ � *η Q η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�*p*�<sup>1</sup> Γð Þ *β* � *p ds* þ *A*1 6j j *λ* þ *γ* � 1 ð *η* 0 ð Þ *<sup>η</sup>* � *<sup>s</sup> <sup>β</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>β</sup> ds* 9 = ; þ*L*1ð Þ k k *u*<sup>2</sup> � *u*<sup>1</sup> þ k k *v*<sup>2</sup> � *v*<sup>1</sup> ð*t* 0 ð Þ *<sup>t</sup>* � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 8 < : þ *MγB*<sup>1</sup> 6 1ð Þ � *η Q* ð 1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�*p*�<sup>1</sup> Γð Þ *α* � *p ds* þ *A*1*γ Q*j j *λ* þ *γ* � 1 ð 1 0 ð Þ <sup>1</sup> � *<sup>s</sup> <sup>α</sup>*�<sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup> ds* 9 = ; ≤ð Þ *L*2*Δ*<sup>1</sup> þ *L*1*Δ*<sup>2</sup> ð Þ k k *u*<sup>2</sup> � *u*<sup>1</sup> þ k k *v*<sup>2</sup> � *v*<sup>1</sup> *:* (54)

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

Thus

$$\left| \left| T\_1(\boldsymbol{u}\_2, \boldsymbol{v}\_2) - T\_1(\boldsymbol{u}\_1, \boldsymbol{v}\_1) \right| \right| \le (L\_2 \Delta\_1 + L\_1 \Delta\_2)(\left| \left| \boldsymbol{u}\_2 - \boldsymbol{u}\_1 \right| + \left| \left| \boldsymbol{v}\_2 - \boldsymbol{v}\_1 \right| \right|). \tag{55}$$

Similarly,

$$\left| \left| T\_2(u\_2, v\_2) - T\_2(u\_1, v\_1) \right| \right| \le (L\_2 \Delta\_3 + L\_1 \Delta\_4) (\left| \left| u\_2 - u\_1 \right| \right| + \left| \left| v\_2 - v\_1 \right| \right|). \tag{56}$$

It follows from (101) and (110) that

$$\left\|\left|T(\boldsymbol{u}\_{2},\boldsymbol{v}\_{2})-T(\boldsymbol{u}\_{1},\boldsymbol{v}\_{1})\right\|\right\| \leq \left(L\_{2}(\Delta\_{1}+\Delta\_{3})+L\_{1}(\Delta\_{2}+\Delta\_{4})\right)(\left\|\boldsymbol{u}\_{2}-\boldsymbol{u}\_{1}\right\|+\left\|\boldsymbol{v}\_{2}-\boldsymbol{v}\_{1}\right\|). \tag{57}$$

Since *L*2ð Þþ *Δ*<sup>1</sup> þ *Δ*<sup>3</sup> *L*1ð Þ *Δ*<sup>2</sup> þ *Δ*<sup>4</sup> < 1, thus *T* is a contraction operator. Hence it follows by Banach's contraction principle that the boundary value problem (1) has a unique solution on 0, 1 ½ �.

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

$$\begin{cases} \,^c D^3 u(t) = \frac{t}{8} \left( (\cos(t)) \sin\left(\frac{|u(t)| + |v(t)|}{2}\right) \right) + \frac{e^{-\left(u(t) + v(t)\right)^2}}{\mathbf{1} + t^2}, & t \in [0, 1], \\\\ \,^c D^3 v(t) = \frac{1}{32} \sin\left(2\pi u(t)\right) + \frac{|v(t)|}{\mathbf{1}6(\mathbf{1} + |v(t)|)} + \frac{\mathbf{1}}{2}, & t \in [0, 1], \end{cases} \tag{58}$$

subject to the three-point coupled boundary conditions

$$\begin{cases} \frac{1}{100}u(0) + \frac{1}{10}u(1) = u\left(\frac{1}{2}\right), \\\\ u(0) = \int\_0^{0,5} u(s)ds, \\\\ \frac{1}{100}D^2u(0) + \frac{1}{10}D^2u(1) = ^cD^2u\left(\frac{1}{2}\right), \end{cases} \tag{59}$$

where *f t*ð Þ¼ , *<sup>u</sup>*, *<sup>v</sup> <sup>t</sup>* <sup>8</sup> ð Þ cosð Þ*<sup>t</sup>* sin j jþ*<sup>u</sup>* j j *<sup>v</sup>* 2 � � � � <sup>þ</sup> *<sup>e</sup>*�ð Þ *<sup>u</sup>*þ*<sup>v</sup>* <sup>2</sup> <sup>1</sup>þ*t*<sup>2</sup> , *<sup>t</sup>*∈½ � 0, 1 , *<sup>η</sup>* <sup>¼</sup> 0, 5, *<sup>λ</sup>* <sup>¼</sup> 0, 01, *<sup>γ</sup>* <sup>¼</sup> 0, 1, *<sup>p</sup>* <sup>¼</sup> 1, 5 *and g t*ð Þ¼ , *<sup>u</sup>*, *<sup>v</sup>* <sup>1</sup> <sup>32</sup>*<sup>π</sup>* sin 2ð Þþ *<sup>π</sup>u t*ð Þ j j *v t*ð Þ 16 1ð Þ þj j *v t*ð Þ <sup>þ</sup> <sup>1</sup> 2 . It can be easily found that *<sup>M</sup>* <sup>¼</sup> <sup>20</sup> <sup>3</sup> and *<sup>Q</sup>* <sup>¼</sup> <sup>9</sup> <sup>400</sup>*.*

Furthermore, by simple computation, for every *ui*, *vi* ∈ , *i* ¼ 1, 2, we have

$$|f(t, u\_1, u\_2) - f(t, v\_1, v\_2)| \le L(|u\_1 - v\_1| + |u\_2 - v\_2|),\tag{60}$$

and

$$|\mathbf{g}(t, u\_1, u\_2) - \mathbf{g}(t, v\_1, v\_2)| \le L(|u\_1 - v\_1| + |u\_2 - v\_2|),\tag{61}$$

where *<sup>L</sup>*<sup>1</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> <sup>¼</sup> *<sup>L</sup>* <sup>¼</sup> <sup>1</sup> <sup>16</sup>. It can be easily found that Δ<sup>1</sup> ¼ *Δ*<sup>3</sup> ffi 0, 799562, Δ<sup>2</sup> ¼ *Δ*<sup>4</sup> ffi 1, 182808.

Finally, since *L*1ð Þþ *Δ*<sup>1</sup> þ *Δ*<sup>3</sup> *L*2ð Þ¼ *Δ*<sup>2</sup> þ *Δ*<sup>4</sup> 2*L*ð Þffi *Δ*<sup>1</sup> þ *Δ*<sup>2</sup> 0, 247796<1, thus all assumptions and conditions of Theorem 1.2 are satisfied. Hence, Theorem implies that the three-point boundary value problem (58, 59) has a unique solution.

#### **3. Existence result for second problem (2)**

We provide another results about the existence of solutions for the problem (2) by using the assumption.

We shall consider the Banach space *B* ¼ *C*ð Þ ½ � 0, 1 equipped with usual supermum norm and *B*<sup>þ</sup> ¼ *C*þð Þ ½ � 0, 1 . In arriving our results, we present some notation and preliminary lemmas. The first is well known.

Lemma 3.1. *Let y t*ð Þ∈*C*½ � 0, 1 *. If u*∈*C*4½ � 0, 1 *, then the BVP*

$$\begin{cases} \ u^{(4)}(t) = \mathfrak{y}(t), & 0 \le t \le 1, \\\ u(0) = \mathfrak{u}'(0) = \mathfrak{u}''(1) = \mathfrak{u}^{\prime\prime}(1) = 0, \end{cases} \tag{62}$$

has a unique solution

$$u(t) = \int\_0^1 G(t, s)\chi(s)ds,\tag{63}$$

where

$$G(t,s) = \begin{cases} \frac{1}{6}t^2(3s-t), & 0 \le t \le s \le 1, \\\frac{1}{6}s^2(3t-s), & 0 \le s \le t \le 1. \end{cases} \tag{64}$$

Lemma 3.2. *For any t*ð Þ , *s* ∈ ½ �� 0, 1 ½ � 0, 1 *, we have*

$$\mathbf{0} \le G(t, s) \le G(\mathbf{1}, s) = \frac{1}{\mathbf{6}} \mathbf{s}^2(\mathbf{3} - s) = \boldsymbol{\nu}(\mathbf{s}).\tag{65}$$

Proof. The derivatives of the function *G* with respect to *t* is

$$\frac{\partial}{\partial t}G(t,s) = \begin{cases} \frac{1}{2}s^2 - \frac{1}{2}(s-t)^2, & 0 \le t \le s \le 1\\ 1, & 0 \le s \le t \le 1. \end{cases} \tag{66}$$

Since the derivative of the function *G* with respect to *t* is nonnegative for all *t*∈½ � 0, 1 , *G* is nondecreasing function of *t* that attaints its maximum when *t* ¼ 1. Then

$$\max\_{0 \le t \le 1} G(t, s) = G(\mathbf{1}, s) = \frac{1}{2}\mathfrak{s}^2 - \frac{1}{6}\mathfrak{s}^3. \tag{67}$$

Lemma 3.3. *Let* 0<*θ* <1*. Then for y t*ð Þ∈*C*þ½ � 0, 1 *, the unique solution u t*ð Þ *of BVP* ð Þ 14 *is nonnegative and satisfies*

$$\min\_{u \in [\theta, 1]} u(t) \ge \frac{2\theta^3}{3} ||u||. \tag{68}$$

Proof. Let *y t*ð Þ∈*C*þ½ � 0, 1 , then from *G t*ð Þ , *s* ≥ 0 we know *u* ∈*C*þ½ � 0, 1 . Set *u t*ð Þ¼ <sup>0</sup> k k*u* , *t*<sup>0</sup> ∈ð � 0, 1 . We first prove that

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

$$\frac{G(t,s)}{G(t\_0,s)} \ge \frac{2}{3}t^3, \quad t, t\_0, s \in (0, 1]. \tag{69}$$

In fact, we can consider four cases:

1.if 0< *t*, *t*<sup>0</sup> ≤*s* ≤1, then

$$\frac{G(t,s)}{G(t\_0,s)} = \frac{t^2(3s-t)}{t\_0^2(3s-t\_0)} \ge \frac{t^2(2s)}{3s-t\_0} \ge \frac{t^2(2s)}{3} \ge \frac{t^2(2t)}{3} = \frac{2t^3}{3},\tag{70}$$

2.if 0< *t*≤*s*≤ *t*<sup>0</sup> ≤1, then

$$\frac{G(t,s)}{G(t\_0,s)} = \frac{t^2(3s-t)}{t\_0^2(3s-t\_0)} \ge \frac{t^2(2s)}{3s-t\_0} \ge \frac{t^2(2s)}{3} \ge \frac{t^2(2t)}{3} = \frac{2t^3}{3},\tag{71}$$

3.if 0< *s*≤*t*, *t*<sup>0</sup> ≤1, then

$$\frac{G(t,s)}{G(t\_0,s)} = \frac{s^2(3t-s)}{s^2(3t\_0-s)} = \frac{3t-s}{3t\_0-s} \ge \frac{3t-s}{3t\_0} \ge \frac{3t-s}{3} \ge \frac{2t+t-s}{3} \ge \frac{2t}{3} \ge \frac{2t^3}{3},\tag{72}$$

4.if 0< *t*<sup>0</sup> ≤*s*≤*t* ≤1, then

$$\frac{G(t,s)}{G(t\_0,s)} = \frac{s^2(3t-s)}{t\_0^2(3t-t\_0)} \ge \frac{t\_0^2(3t-s)}{t\_0^2(3t-t\_0)} \ge \frac{3t-s}{3t} \ge \frac{3t-t}{3t} \ge \frac{2t}{3} \ge \frac{2t^3}{3},\tag{73}$$

Therefore, for *t*∈ ½ � *θ*, 1 , we have

$$u(t) = \int\_0^1 G(t, s)\boldsymbol{\eta}(s)ds = \left[\frac{G(t, s)}{G(t\_0, s)}G(t\_0, s)\boldsymbol{\eta}(s)ds \ge \frac{2t^3}{3}\boldsymbol{u}(t\_0) \ge \frac{2\theta^3}{3}||\boldsymbol{u}||.\tag{74}$$

The proof is complete. If we let

$$K = \left\{ \mathbf{x} \in B : \begin{array}{ll} \mathbf{x}(t) \ge \mathbf{0} \text{ on } \ [0, 1], \text{ and } \min\_{t \in [\theta, 1]} \mathbf{x}(t) \ge \frac{2\theta^3}{3} \|\mathbf{x}\| \right\} \end{array} \tag{75}$$

then it is easy to see that *K* a cone in *B*. We not that a pair ð Þ *u t*ð Þ, *v t*ð Þ is a solution of BVPs (2) if, and only if

$$u(t) = \lambda \int\_0^1 G(t, s) a(s) f\left(\lambda \int\_0^1 G(s, r) b(r) g(u(r)) dr\right) ds, \quad t \in [0, 1],\tag{76}$$

and

$$v(t) = \lambda \int\_0^1 G(t, s) b(s) g(u(s)) ds, \quad t \in [0, 1]. \tag{77}$$

Now, we define an integral operator *T* : *K* ! *B* by

$$\mathbf{u}(Tu)(t) = \lambda \int\_{0}^{1} G(t,s)a(s)f\left(\lambda \int\_{0}^{1} G(s,r)b(r)\mathbf{g}(u(r))dr\right)ds, \quad u \in K. \tag{78}$$

We adopt the following assumptions:

ð Þ *H*<sup>1</sup> *a*, *b*∈*C*ð Þ ð Þ 0, 1 , 0, ½ Þ ∞ and each does not vanish identically on any subinterval. ð Þ *H*<sup>2</sup> *f*, *g* ∈*C*ð Þ ½ Þ 0, ∞ , 0, ½ Þ ∞ and each to be singular at *t* ¼ 0*or t* ¼ 1.

ð Þ *H*<sup>3</sup> All of *f* <sup>0</sup> ¼ lim *x*!0<sup>þ</sup> *f x*ð Þ *<sup>x</sup>* , *g*<sup>0</sup> ¼ lim *x*!0<sup>þ</sup> *g x*ð Þ *<sup>x</sup>* , *<sup>f</sup>* <sup>∞</sup> <sup>¼</sup> lim*x*!<sup>∞</sup> *f x*ð Þ *<sup>x</sup>* , *and g*<sup>∞</sup> <sup>¼</sup> lim*x*!<sup>∞</sup> *g x*ð Þ *<sup>x</sup>* exist

as real numbers.

ð Þ *H*<sup>4</sup> *g*ð Þ¼ 0 0 and *f* is increasing function.

Lemma 3.4 *Let λ be positive number and K be the cone defined above.*

i. If *u*∈ *B*<sup>þ</sup> and *v* : ½ �! 0, 1 ½ Þ 0, ∞ is defined by (77), then *v*∈*K*.


Proof. Let *u*∈*B*<sup>þ</sup> and *v* be defined by (77).

i. By the nonnegativity of *G*, *b* and *g* it follows that *v t*ð Þ≥ 0, *t*∈ ½ � 0, 1 . In view of ð Þ *H*<sup>1</sup> ,ð Þ *H*<sup>2</sup> , we have

$$\int\_{0}^{1} G(t,s)b(s)\mathfrak{g}(u(s))ds \geq \int\_{t}^{1} \min\_{t \in [\theta,1]} G(t,s)b(s)\mathfrak{g}(u(s))ds,\tag{79}$$

from which, we take

$$\min\_{t \in [\theta, 1]} \int\_0^1 G(t, s) b(s) \mathbf{g}(u(s)) ds \ge \int\_{t \in [\theta, 1]}^1 \min\_{t \in [\theta, 1]} G(t, s) b(s) \mathbf{g}(u(s)) ds. \tag{80}$$

Consequently, employing (68) and for *λ*>0, we have

$$\begin{split} \lambda \int\_{0}^{1} G(t,s)b(s)\mathbf{g}(u(s))ds &\geq \lambda \int\_{0}^{1} \min\_{t\in[\theta,1]} G(t,s)b(s)\mathbf{g}(u(s))ds \\ &\geq \frac{2\theta^{3}}{3}\lambda \int\_{0}^{1} G(t\_{0},s)b(s)\mathbf{g}(u(s))ds \\ &\geq \frac{2\theta^{3}}{3}v(t\_{0}), t\_{0}\in(0,1] \\ &\geq \frac{2\theta^{3}}{3}||v||. \end{split} \tag{81}$$

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

Therefore

$$\min\_{t \in [\theta, 1]} v(t) \ge \frac{2\theta^3}{3} ||v||. \tag{82}$$

Which give that *v*∈ *K*.

ii. Obviously, for *v*∈ *K*, *T u*ð Þ∈*C*þ½ � 0, 1 . For *t*∈ ½ � 0, 1 , we have

$$\begin{aligned} \|Tu(t)\| &= \max\_{0 \le t \le 1} \lambda \int\_0^1 G(t,s)a(s)f(v(s))ds\\ &\le \lambda \int\_0^1 G(1,s)a(s)f(v(s))ds, \end{aligned} \tag{83}$$

and

$$\begin{split} Tu(t) &= \lambda \int\_0^1 G(t,s)a(s)f(v(s))ds \\ &= \lambda \int\_0^1 \frac{G(t,s)}{G(1,s)}G(1,s)a(s)f(v(s))ds \\ &\geq \frac{2\theta^3}{3}\lambda \int\_0^1 G(1,s)a(s)f(v(s))ds \\ &\geq \frac{2}{3}\theta^3||Tu(t)||. \end{split} \tag{84}$$

Which give that *Tu*∈*K*. Therefore *T* : *K* ! *K*.

iii. By using standard arguments it is not difficult to show that the operator *T* : *K* ! *B* is completely continuous.

The key tool in our approach is the following Krasnoselskii's fixed point theorem of cone expansion-compression type.

Theorem 1.3 (See [47]) Let *B* be a Banach space and *K* ⊂*B* be a cone in *B*. Assume Ω<sup>1</sup> and Ω<sup>2</sup> are open subset of *B* with 0 ∈ Ω<sup>1</sup> and Ω<sup>1</sup> ⊂ Ω2,

*T* : *K* ∩ Ω2nΩ<sup>1</sup> � � ! *<sup>K</sup>* be a completely continuous operator such that.


Then, *T* has a fixed point in *K* ∩ Ω2nΩ<sup>1</sup> � �. Throughout this section, we shall use the following notations:

*Boundary Layer Flows - Modelling, Computation, and Applications of Laminar,Turbulent…*

$$L\_1 = \max\left\{ \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_\theta^1 \nu(r) a(r) f\_\omega dr \right\}^{-1}, \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_\theta^1 \nu(r) a(r) g\_\omega dr \right\}^{-1} \tag{85}$$

and

$$L\_2 = \min \left\{ \left[ \int\_0^1 \wp(r) a(r) f\_0 dr \right]^{-1}, \left[ \int\_0^1 \wp(r) b(r) g\_0 dr \right]^{-1} \right\}. \tag{86}$$

$$L\_3 = \max\left\{ \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_\theta^1 \mu(r) a(r) f\_0 dr \right\}^{-1}, \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_\theta^1 \nu(r) a(r) \mathbf{g}\_0 dr \right\}^{-1} \tag{87}$$

and

$$L\_4 = \min \left\{ \left[ \int\_{\theta}^1 \mu(r) a(r) f\_{\bowtie} dr \right]^{-1}, \left[ \int\_{\theta} \mu(r) b(r) g\_{\bowtie} dr \right]^{-1} \right\}. \tag{88}$$

#### **4. Existence results**

In this section, we discuss the existence of at least one positive solution for BVPs (2). We obtain the following existence results, by applying the positivity of Green's function *G t*ð Þ , *s* and the fixed-point of cone expansion-compression type.

Theorem 1.4 Assume conditions ð Þ *H*<sup>1</sup> ,ð Þ *H*<sup>2</sup> and ð Þ *H*<sup>3</sup> are satisfied. Then, for each *λ* satisfying *L*<sup>1</sup> <*λ*< *L*<sup>2</sup> there exists a pair ð Þ *u*, *v* satisfying BVPs (2) such that *u t*ð Þ>0 and *v t*ð Þ>0 on 0, 1 ð Þ.

Proof. Let *L*<sup>1</sup> < *λ*<*L*2. And let *ε*> 0 be chosen such that

$$\max \left\{ \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_{\theta}^1 \left[ \varphi(r)a(r)(f\_{\text{os}} - \epsilon) dr \right]^{-1}, \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_{\theta}^1 \left[ \varphi(r)a(r)(g\_{\text{os}} - \epsilon) dr \right]^{-1} \right\} \le \lambda, \tag{89}$$

and

$$\lambda \le \min \left\{ \left[ \int\_{\theta}^{1} \wp(r) a(r) (f\_0 + \varepsilon) dr \right]^{-1}, \left[ \int\_{\theta}^{1} \wp(r) b(r) (g\_0 + \varepsilon) dr \right]^{-1} \right\}. \tag{90}$$

From the definitions of *f* <sup>0</sup> and *g*<sup>0</sup> there exists an *R*<sup>1</sup> >0 such that

$$f(u) \le (f\_0 + \varepsilon)u, \quad 0 < u \le R\_1,\tag{91}$$

and

$$\mathbf{g}\left(\boldsymbol{u}\right) \le \left(\mathbf{g}\_0 + \varepsilon\right)\boldsymbol{u}, \ \mathbf{0} < \boldsymbol{u} \le \boldsymbol{R}\_1,\tag{92}$$

Let *u*∈ *K* with k k*u* ¼ *R*1. From (65) and choice of *ε*, we have

$$\begin{split} \lambda \int\_{0}^{1} G(t,s)b(r)\mathfrak{g}(u(r)) &\leq \lambda \int\_{0}^{1} \mathfrak{w}(r)b(r)\mathfrak{g}(u(r))dr \\ &\leq \lambda \int\_{0}^{1} \mathfrak{w}(r)b(r)\left(\mathfrak{g}\_{0} + \varepsilon\right)u(r)dr \\ &\leq ||u||\lambda \int\_{0}^{1} \mathfrak{w}(r)b(r)dr(\mathfrak{g}\_{0} + \varepsilon)dr \\ &\leq R\_{1} = ||u||. \end{split} \tag{93}$$

Consequently, from (65) and choice of *ε*, we have

$$\begin{split} T u(t) &= \lambda \int\_{0}^{1} G(t,s) a(s) f' \left( \lambda \int\_{0}^{1} G(s,r) b(r) g(u(r)) dr \right) ds \\ &\leq \lambda \int\_{0}^{1} \varphi(s) a(s) f' \left( \lambda \int\_{0}^{1} G(s,r) b(r) g(u(r)) dr \right) ds \\ &\leq \lambda \int\_{0}^{1} \varphi(s) a(s) \left( f\_{0} + \varepsilon \right) \left[ \lambda \int\_{0}^{1} G(s,r) b(r) g(u(r)) dr \right] ds \\ &\leq \lambda \int\_{0}^{1} \varphi(s) a(s) \left( f\_{0} + \varepsilon \right) R\_{1} ds \\ &\leq R\_{1} = ||u||. \end{split} \tag{94}$$

So, k k *Tu* ≤k k*u* . If we set Ω<sup>1</sup> ¼ f g *u* ∈*B* : k k*u* < *R*<sup>1</sup> , then

$$||Tu|| \le ||u||, \text{ for } u \in K \cap \partial\Omega\_1 \tag{95}$$

Considering the definitions of *f* <sup>∞</sup> and *g*<sup>∞</sup> there exists an *R*<sup>2</sup> >0 such that

$$f(u) \ge (f\_{\ll} - \varepsilon)u, \quad 0 < u \le \overline{R}\_2,\tag{96}$$

and

$$\mathcal{g}(u) \ge (\mathcal{g}\_{\infty} - \varepsilon)u, \, 0 < u \le \overline{R}\_2. \tag{97}$$

Let *<sup>u</sup>*<sup>∈</sup> *<sup>K</sup>* and *<sup>R</sup>*<sup>2</sup> <sup>¼</sup> max 2*R*1, <sup>3</sup>*R*<sup>2</sup> 2*θ*<sup>3</sup> n o with k k*<sup>u</sup>* <sup>¼</sup> *<sup>R</sup>*2, then

$$\min\_{s \in [\theta, 1]} u(s) \ge \frac{2}{3} \theta^3 ||u|| \ge \overline{R}\_2 \tag{98}$$

Thus, from (68) and choice of *ε*, we have

$$\begin{split} \lambda \int\_{0}^{1} G(t,s)b(r)\mathbf{g}(u(r)) &\geq \frac{2\theta^{3}}{3} \lambda \int\_{0}^{1} G(1,r)b(r)\mathbf{g}(u(r))dr\\ &\geq \frac{2\theta^{3}}{3} \lambda \int\_{\theta}^{1} \mathbf{y}(r)b(r) \left(\mathbf{g}\_{\infty}-\varepsilon\right)u(r)dr\\ &\geq ||u|| \left(\frac{2\theta^{3}}{3}\right)^{2} \lambda \int\_{\theta}^{1} \mathbf{y}(r)b(r)dr (\mathbf{g}\_{\infty}-\varepsilon)dr\\ &\geq R\_{2} = ||u||. \end{split} \tag{99}$$

Consequently, from (77) and choice of *ε*, we have

$$\begin{split} Tu(t) &\geq \frac{2\theta}{3} \lambda \int\_{0}^{1} \left\Vert \boldsymbol{\nu}(s)\boldsymbol{a}(s)\boldsymbol{f}^{\prime} \left(\lambda \int\_{0}^{1} \boldsymbol{G}(s,r)\boldsymbol{b}(r)\boldsymbol{g}(u(r))dr\right)ds \\ &\geq \frac{2\theta^{3}}{3} \lambda \int\_{0}^{1} \boldsymbol{\nu}(s)\boldsymbol{a}(s) \left(\boldsymbol{f}\_{\infty} - \varepsilon\right) \left[\lambda \int\_{\theta}^{1} \boldsymbol{G}(s,r)\boldsymbol{b}(r)\boldsymbol{g}(u(r))dr\right]ds \\ &\geq \frac{2\theta^{3}}{3} \lambda \int\_{\theta}^{1} \boldsymbol{\nu}(s)\boldsymbol{a}(s) \left(\boldsymbol{f}\_{\infty} - \varepsilon\right)\boldsymbol{H}\_{2}ds \\ &\geq \left(\frac{2\theta^{3}}{3}\right)^{2} \lambda \int\_{0}^{1} \boldsymbol{\nu}(s)\boldsymbol{a}(s) \left(\boldsymbol{f}\_{\infty} - \varepsilon\right)\boldsymbol{H}\_{2}ds \\ &\geq R\_{2} = ||\boldsymbol{u}||. \end{split} \tag{100}$$

So, k k *Tu* ≥k k*u* . If we set Ω<sup>2</sup> ¼ f g *u*∈*B* : k k*u* <*R*<sup>2</sup> , then

$$||Tu|| \ge ||u||, \text{ for } u \in K \cap \partial\Omega\_2. \tag{101}$$

Applying (i) of Theorem 3.1 to (95) and (101), yields that *T* has a fixed point *u*<sup>∗</sup> ∈ *K* ∩ Ω2*=*Ω<sup>1</sup> � �. As such and with *v* defined by

$$v(t) = \lambda \int\_0^1 \mathbf{G}(t, s) b(s) \mathbf{g}(u(s)) ds,\tag{102}$$

the pair ð Þ *u*, *v* is a desired solution of BVPs (2) for the given *λ*. The proof is complete. Theorem 1.5 Assume conditions ð Þ *H*<sup>1</sup> ,ð Þ *H*<sup>2</sup> ,ð Þ *H*<sup>3</sup> and ð Þ *H*<sup>4</sup> are satisfied. Then, for each *λ* satisfying *L*<sup>3</sup> <*λ*< *L*<sup>4</sup> there exists a pair ð Þ *u*, *v* satisfying BVPs (2) such that *u t*ð Þ>0 and *v t*ð Þ>0 on 0, 1 ð Þ.

Proof. Let *L*<sup>3</sup> < *λ*<*L*<sup>4</sup> and *ε* >0 be chosen such that

$$\max \left\{ \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_{\theta}^1 \left[ \psi(r)a(r)(f\_0 - \varepsilon)dr \right]^{-1}, \left[ \left( \frac{2\theta^3}{3} \right)^2 \right]\_{\theta}^1 \left[ \psi(r)a(r)(g\_0 - \varepsilon)dr \right]^{-1} \right\} \le \lambda, \tag{103}$$

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

and

$$\lambda \le \min \left\{ \left[ \int\_0^1 \wp(r) a(r) (f\_{\infty} + \varepsilon) dr \right]^{-1}, \left[ \int\_0^1 \wp(r) b(r) (\mathfrak{g}\_{\infty} + \varepsilon) dr \right]^{-1} \right\}. \tag{104}$$

From the definitions of *f* <sup>0</sup> and *g*<sup>0</sup> there exists an *R*<sup>1</sup> >0 such that

$$f(u) \ge (f\_0 - \varepsilon)u, \quad 0 < u \le R\_1,\tag{105}$$

and

$$\mathbf{g}(u) \ge (\mathbf{g}\_0 - \varepsilon)u, \quad \mathbf{0} < u \le \mathbf{R}\_1,\tag{106}$$

Now *g*ð Þ¼ 0 0 and so there exists 0< *R*<sup>2</sup> ≤ *R*<sup>1</sup> such that

$$\lambda \mathbf{g}(u) \le \frac{R\_1}{\frac{1}{2}}, \quad 0 \le u \le R\_2. \tag{107}$$

$$\int\_0 \mathbf{y}(r) b(r) dr$$

Let *u*∈ *K* with k k*u* ¼ *R*2. Then

$$\begin{aligned} \lambda \int\_0^1 G(t,s)b(r)g(u(r)) &\le \lambda \int\_0^1 \mu(r)b(r)g(u(r))dr\\ &\le \begin{cases} \int \mu(r)b(r)R\_1 dr\\ \frac{0}{1} \end{cases} \\ &\le \frac{1}{\int\_0^1 \mu(s)b(s)ds} \\ &\le R\_1 = ||u||. \end{aligned} \tag{108}$$

Therefore, by (68), we have

$$\begin{split} Tu(t) &= \lambda \int\_{0}^{1} G(t,s)a(s)\xi'\left(\lambda \prod\_{o=1}^{1} G(s,r)b(r)\lg(u(r))dr\right)ds \\ &\geq \frac{2\theta^{3}}{3}\lambda \int\_{0}^{1} \wp(s)a(s)\xi'\left(\frac{2\theta^{3}}{3}\lambda \prod\_{o=1}^{1} \wp(r)b(r)\lg(u(r))dr\right)ds \\ &\geq \frac{2\theta^{3}}{3}\lambda \int\_{\theta}^{1} \wp(s)a(s)\left(\mathcal{f}\_{0}-\epsilon\right)\left[\left(\frac{2\theta^{3}}{3}\right)^{2}\lambda\right]\Big{|}\varkappa(r)b(r)\left(\mathcal{g}\_{0}-\epsilon\right)||\mathbbm{u}||dr \\ &\geq \frac{2\theta^{3}}{3}\lambda \int\_{\theta}^{1} \wp(r)a(r)\left(\mathcal{f}\_{0}-\epsilon\right)||\mathbbm{u}|| \\ &\geq \left(\frac{2\theta^{3}}{3}\right)^{2}\lambda \int\_{\theta}^{1} \wp(r)a(r)\left(\mathcal{f}\_{0}-\epsilon\right)||\mathbbm{u}|| \\ &\geq ||\mathbbm{u}||. \end{split} \tag{109}$$

So, k k *Tu* ≥k k*u* . If we set Ω<sup>1</sup> ¼ f g *u* ∈*B* : k k*u* < *R*<sup>2</sup> , then

$$||Tu|| \ge ||u||, \quad u \in K \cap \left(\overline{\Omega}\_2 \backslash \Omega\_1\right) \tag{110}$$

Considering the definitions of *f* <sup>∞</sup> and *g*<sup>∞</sup> there exists *R*<sup>1</sup> >0 such that

$$f(u) \le (f\_{\ll} + \varepsilon)u, \quad u \ge \overline{R}\_1,\tag{111}$$

and

$$\mathbf{g}(u) \le (\mathbf{g}\_{\infty} + \varepsilon)u, \quad u \ge \overline{R}\_1. \tag{112}$$

We consider two cases: *g* is bounded or *g* is unbounded.

Case ð Þ*i* . Suppose *g* is bounded, say *g u*ð Þ≤ *N*, *N* >0 for all 0 <*u* < ∞. Then, for *u*∈ *K*

$$\begin{aligned} \lambda \int\_0^1 \mathbf{G}(t, s) b(r) \mathbf{g}(u(r)) &\le \lambda \int\_0^1 \mu(r) b(r) \mathbf{g}(u(r)) dr \\ \mathcal{M} = \max \left\{ f(u) : \ 0 \le u \le N \lambda \int\_0^1 \mu(r) b(r) dr \right\} \end{aligned} \tag{113}$$

and let

$$R\_3 > \max\left\{2R\_2, \ M\lambda \left|\psi(s)a(s)ds\right|\right\}.\tag{114}$$

Then, for *u* ∈*K* with k k*u* ¼ *R*3, we have

$$\begin{aligned} Tu(t) &\le \lambda \int\_0^1 \mu(s) a(s) M ds \\ &\le R\_3 = ||u||. \end{aligned} \tag{115}$$

So that k k *Tu* <sup>≤</sup> k k*<sup>u</sup>* . If we set <sup>Ω</sup><sup>2</sup> <sup>¼</sup> f g *<sup>u</sup>* <sup>∈</sup>*<sup>B</sup>* : k k*<sup>u</sup>* <sup>≤</sup> *<sup>R</sup>*<sup>3</sup> , then, for *<sup>u</sup>* <sup>∈</sup>*<sup>K</sup>* <sup>∩</sup> *<sup>∂</sup>*Ω2:

$$||Tu|| \le ||u||, \quad u \in K \cap \partial \Omega\_2 \tag{116}$$

Caseð Þ *ii* . *g* is unbounded, there exists *R*<sup>3</sup> > *max* 2*R*2, *R*<sup>1</sup> � � such that *g u*ð Þ≤*g R*ð Þ<sup>3</sup> , for 0<*u*≤ *R*3.

Similarly, there exists *R*<sup>4</sup> > *max* 2*R*3, *Mλ* Ð 1 <sup>0</sup>*ψ*ð Þ*<sup>r</sup> b r*ð Þ*g R*ð Þ<sup>3</sup> *ds* n o such that *f u*ð Þ≤*f R*ð Þ<sup>4</sup> , for 0< *u*≤*R*4.

*A Study for Coupled Systems of Nonlinear Boundary Value Problem DOI: http://dx.doi.org/10.5772/intechopen.105428*

Let *u*∈ *K* with k k*u* ¼ *R*4, from ð Þ *H*<sup>4</sup> , we have

$$\begin{split} T u(t) &\leq \lambda \prod\_{0}^{1} \varphi(s) a(s) f\left(\lambda \prod\_{0}^{1} \varphi(r) b(r) g(R\_{3}) dr\right) ds \\ &\leq \lambda \int\_{0}^{1} \varphi(r) a(r) f(R\_{4}) ds \\ &\leq \lambda \int\_{0}^{1} \varphi(r) a(r) \left(f|\_{\infty} + \varepsilon\right) R\_{4} ds \\ &\leq R\_{4} = ||u||. \end{split} \tag{117}$$

So, k k *Tu* ≤k k*u* . If we set Ω<sup>2</sup> ¼ f g *x*∈*C*½ �j 0, 1 k k*x* ≤*R*<sup>4</sup> , then

$$||Tu|| \le ||u||, \quad for \quad u \in K \cap \partial\Omega\_2. \tag{118}$$

In either of cases, application of part ð Þ *ii* of Theorem 3.1 yields a fixed point *<sup>u</sup>*<sup>∗</sup> of *T* belonging to *K* ∩ Ω2*=*Ω<sup>1</sup> � �, which in turn yields a pair ð Þ *<sup>u</sup>*, *<sup>v</sup>* satisfying BVPs (2) for the chosen value of *λ*. The proof is complete.

We construct an example to illustrate the applicability of the results presented. Example 4.1. *Consider the two-point boundary value problem*

$$\begin{cases} u^{(4)}(t) = \lambda t v(t) \left( v(t) e^{-v(t)} + \frac{v(t) + K}{\mathbf{1} + \eta v(t)} \right), \quad 0 < t < 1, \\\\ v^{(4)}(t) = \lambda t u(t) \left( u(t) e^{-u(t)} + \frac{u(t) + K}{\mathbf{1} + \eta u(t)} \right), \quad 0 < t < 1, \end{cases} \tag{119}$$

and satisfying two-point boundary conditions

$$\begin{cases} \boldsymbol{u}(\mathbf{0}) = \mathbf{0}, \boldsymbol{u}'(\mathbf{0}) = \mathbf{0}, \boldsymbol{u}''(\mathbf{1}) = \mathbf{0}, \boldsymbol{u}'''(\mathbf{1}) = \mathbf{0}, \\\ \boldsymbol{v}(\mathbf{0}) = \mathbf{0}, \boldsymbol{v}'(\mathbf{0}) = \mathbf{0}, \boldsymbol{v}''(\mathbf{1}) = \mathbf{0}, \boldsymbol{v}'''(\mathbf{1}) = \mathbf{0}, \end{cases} \tag{120}$$

where *a t*ðÞ¼ *b t*ðÞ¼ *<sup>t</sup>*, *f v*ð Þ¼ *v ve*�*<sup>v</sup>* <sup>þ</sup> *<sup>v</sup>*þ*<sup>K</sup>* 1þ*ηv* � �, *g u*ð Þ¼ *<sup>u</sup>* <sup>1</sup> <sup>þ</sup> *<sup>u</sup>*þ*<sup>K</sup>* 1þ*ηu* � �. By simple calculations, we find *<sup>g</sup>*ð Þ¼ <sup>0</sup> 0, *<sup>f</sup>* <sup>∞</sup> <sup>¼</sup> *<sup>g</sup>*<sup>∞</sup> <sup>¼</sup> <sup>1</sup> *<sup>η</sup>* , *f* <sup>0</sup> ¼ *g*<sup>0</sup> ¼ *K.*

Choosing *<sup>θ</sup>* <sup>¼</sup> <sup>1</sup> <sup>3</sup> , *<sup>η</sup>* <sup>¼</sup> 100, *andK* <sup>¼</sup> 104*, we obtain L*<sup>3</sup> ffi 1, 1817237, *<sup>L</sup>*<sup>4</sup> ffi 9, 1666667*.* By Theorem 4, *it follows that for every λ such that* 1, 1817237 < *λ*<9, 1666667*, there exists a pair u*ð Þ , *v satisfying* BVPs (25–2526)*.*

#### **5. Conclusions**

This chapter concerns the boundary value problem of a class of fractional differential equations involving the Caputo fractional derivative with nonlocal boundary conditions. By using the Leray-Shauder nonlinear alternative and Banach contraction principle, one shows that the problem has at least one positive solutions and has unique solution. Secondly, we derive explicit eigenvalue intervals of *λ* for the existence of at least one positive solution for the second problem by using Krasnosel'skii fixed point theorem. The results of the present chapter are significantly contribute to the existing literature on the topic.
