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# A coupled system of fractional differential equations with nonlocal integral boundary conditions

*Advances in Difference Equations*
**volume 2012**, Article number: 130 (2012)

## Abstract

In this paper, we prove the existence and uniqueness of solutions for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach’s fixed-point theorem. An illustrative example is also presented.

**MSC:**34A08, 34A12, 34B15.

## 1 Introduction

In this paper, we investigate a boundary value problem of first-order fractional differential equations with Riemann-Liouville integral boundary conditions of different order given by

where ${}^{c}D_{0+}^{\alpha}$, ${}^{c}D_{0+}^{\beta}$ denote the Caputo fractional derivatives, $0<\alpha ,\beta \le 1$, $f,g\in C([0,1]\times {\mathbb{R}}^{2},\mathbb{R})$, and $p,q,\gamma ,\delta \in \mathbb{R}$.

Fractional differential equations have recently been addressed by several researchers for a variety of problems. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, *etc.* [1–5]. Fractional-order differential equations are also regarded as a better tool for the description of hereditary properties of various materials and processes than the corresponding integer order differential equations. With this advantage, fractional-order models become more realistic and practical than the classical integer-order models, in which such effects are not taken into account. For some recent development on the topic, see [6–18], and the references therein. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications. The reader is referred to the papers [19–22], and the references cited therein.

This paper is organized as follows: In Sect. 2, we present some basic materials needed to prove our main results. In Sect. 3, we prove the existence and uniqueness of solutions for the system (1.1) by applying some standard fixed-point principles.

## 2 Preliminaries

Let us introduce the space $X=\{u(t)|u(t)\in {C}^{1}([0,1])\}$ endowed with the norm $\parallel u\parallel =max\{|u(t)|,t\in [0,1]\}$. Obviously, $(X,\parallel \cdot \parallel )$ is a Banach space. Also, let $Y=\{v(t)|v(t)\in {C}^{1}([0,1])\}$ endowed with the norm $\parallel v\parallel =max\{|v(t)|,t\in [0,1]\}$. The product space $(X\times Y,\parallel (u,v)\parallel )$ is also a Banach space with norm $\parallel (u,v)\parallel =\parallel u\parallel +\parallel v\parallel $.

For the convenience of the readers, we now present some useful definitions and fundamental facts of fractional calculus [1, 4].

**Definition 2.1** For at least *n*-times continuously differentiable function $g:[0,\mathrm{\infty})\to \mathbb{R}$, the Caputo derivative of fractional order *q* is defined as

where $[q]$ denotes the integer part of the real number *q*.

**Definition 2.2** The Riemann-Liouville fractional integral of order *q* is defined as

provided the integral exists.

The following lemmas gives some properties of Riemann-Liouville fractional integrals and Caputo fractional derivative [1].

**Lemma 2.3** *Let* $p,q\ge 0$, $f\in {L}_{1}[a,b]$. *Then* ${I}^{p}{I}^{q}f(t)={I}^{p+q}f(t)$ *and* ${}^{c}D^{q}{I}^{q}f(t)=f(t)$, *for all* $t\in [a,b]$.

**Lemma 2.4** *Let* $\beta >\alpha >0$, $f\in {L}_{1}[a,b]$. *Then* ${}^{c}D^{\alpha}{I}^{\beta}f(t)={I}^{\beta -\alpha}f(t)$, *for all* $t\in [a,b]$.

To define the solution of the boundary value problem (1.1), we need the following lemma, which deals with a linear variant of the problem (1.1).

**Lemma 2.5** *Let* $\gamma \ne \frac{\mathrm{\Gamma}(p+1)}{{\eta}^{p}}$. *Then for a given* $g\in C([0,1],\mathbb{R})$, *the solution of the fractional differential equation*

*subject to the boundary condition*

*is given by*

*Proof* For some constant ${c}_{0}\in \mathbb{R}$, we have [1]

Using the Riemann-Liouville integral of order *p* for (2.4), we have

where we have used Lemma 2.3. Using the condition (2.2) in the above expression, we get

Substituting the value of ${c}_{0}$ in (2.4), we obtain (2.3). □

## 3 Main results

For the sake of convenience, we set

and

Define the operator $T:X\times Y\to X\times Y$ by

The first result is based on Leray-Schauder alternative.

**Lemma 3.1** (Leray-Schauder alternative, [23] p.4)

*Let* $F:E\to E$ *be a completely continuous operator* (*i*.*e*., *a map that restricted to any bounded set in* *E* *is compact*). *Let*

*Then either the set* $\mathcal{E}(F)$ *is unbounded*, *or* *F* *has at least one fixed point*.

**Theorem 3.2** *Suppose that* $\gamma \ne \frac{\mathrm{\Gamma}(p+1)}{{\eta}^{p}}$ *and* $\delta \ne \frac{\mathrm{\Gamma}(q+1)}{{\zeta}^{q}}$. *Assume that there exist real constants* ${k}_{i}$, ${\lambda}_{i}\ge 0$ ($i=1,2$) *and* ${k}_{0}>0$, ${\lambda}_{0}>0$ *such that* $\mathrm{\forall}{x}_{i}\in \mathbb{R}$ ($i=1,2$), *we have*

*In addition*, *it is assumed that*

*where* ${M}_{1}$ *and* ${M}_{2}$ *are given by* (3.1) *and* (3.2), *respectively*. *Then the boundary value problem* (1.1) *has at least one solution*.

*Proof* First, we show that the operator $T:X\times Y\to X\times Y$ is completely continuous. By continuity of functions *f* and *g*, the operator *T* is continuous.

Let $\mathrm{\Omega}\subset X\times Y$ be bounded. Then there exist positive constants ${L}_{1}$ and ${L}_{2}$ such that

Then for any $(u,v)\in \mathrm{\Omega}$, we have

Similarly, we get

Thus, it follows from the above inequalities that the operator *T* is uniformly bounded.

Next, we show that *T* is equicontinuous. Let $0\le {t}_{1}\le {t}_{2}\le 1$. Then we have

Analogously, we can obtain

Therefore, the operator $T(u,v)$ is equicontinuous, and thus the operator $T(u,v)$ is completely continuous.

Finally, it will be verified that the set $\mathcal{E}=\{(u,v)\in X\times Y|(u,v)=\lambda T(u,v),0\le \lambda \le 1\}$ is bounded. Let $(u,v)\in \mathcal{E}$, then $(u,v)=\lambda T(u,v)$. For any $t\in [0,1]$, we have

Then

and

Hence, we have

and

which imply that

Consequently,

for any $t\in [0,1]$, where ${M}_{0}$ is defined by (3.3), which proves that $\mathcal{E}$ is bounded. Thus, by Lemma 3.1, the operator *T* has at least one fixed point. Hence, the boundary value problem (1.1) has at least one solution. The proof is complete. □

In the second result, we prove existence and uniqueness of solutions of the boundary value problem (1.1) via Banach’s contraction principle.

**Theorem 3.3** *Assume that* $f,g:[0,1]\times {\mathbb{R}}^{2}\to \mathbb{R}$ *are continuous functions and there exist constants* ${m}_{i}$, ${n}_{i}$, $i=1,2$ *such that for all* $t\in [0,1]$ *and* ${u}_{i},{v}_{i}\in \mathbb{R}$, $i=1,2$,

*and*

*In addition*, *assume that*

*where* ${M}_{1}$ *and* ${M}_{2}$ *are given by* (3.1) *and* (3.2), *respectively*. *Then the boundary value problem* (1.1) *has a unique solution*.

*Proof* Define ${sup}_{t\in [0,1]}f(t,0,0)={N}_{1}<\mathrm{\infty}$ and ${sup}_{t\in [0,1]}g(t,0,0)={N}_{2}<\mathrm{\infty}$ such that

We show that $T{B}_{r}\subset {B}_{r}$, where ${B}_{r}=\{(u,v)\in X\times Y:\parallel (u,v)\parallel \le r\}$.

For $(u,v)\in {B}_{r}$, we have

Hence,

In the same way, we can obtain that

Consequently, $\parallel T(u,v)(t)\parallel \le r$.

Now for $({u}_{2},{v}_{2}),({u}_{1},{v}_{1})\in X\times Y$, and for any $t\in [0,1]$, we get

and consequently we obtain

Similarly,

It follows from (3.4) and (3.5) that

Since ${M}_{1}({m}_{1}+{m}_{2})+{M}_{2}({n}_{1}+{n}_{2})<1$, therefore, *T* is a contraction operator. So, by Banach’s fixed-point theorem, the operator *T* has a unique fixed point, which is the unique solution of problem (1.1). This completes the proof. □

**Example 3.4** Consider the following system of fractional boundary value problem:

Here, $\alpha =1/2$, $\gamma =\sqrt{3}$, $p=3/2$, $\eta =1/3$, $\beta =1/2$, $\delta =\sqrt{2}$, $q=1/2$, $\zeta =1/2$, and $f(t,u,v)=\frac{1}{4{(t+2)}^{2}}\frac{|u|}{1+|u|}+1+\frac{1}{8}{sin}^{2}v$ and $g(t,u,v)=\frac{1}{32\pi}sin(2\pi u)+\frac{|v|}{16(1+|v|)}+\frac{1}{2}$. Note that $\gamma =\sqrt{3}\ne \mathrm{\Gamma}(p+1)/{\eta}^{p}=\mathrm{\Gamma}(5/2)/{(1/3)}^{3/2}$ and $\delta =\sqrt{2}\ne \mathrm{\Gamma}(q+1)/{\zeta}^{q}=\mathrm{\Gamma}(3/2)/{(1/2)}^{1/2}$. Furthermore, $|f(t,{u}_{1},{u}_{2})-f(t,{v}_{1},{v}_{2})|\le \frac{1}{16}|{u}_{1}-{u}_{2}|+\frac{1}{16}|{v}_{1}-{v}_{2}|$, $|g(t,{u}_{1},{u}_{2})-g(t,{v}_{1},{v}_{2})|\le \frac{1}{16}|{u}_{1}-{u}_{2}|+\frac{1}{16}|{v}_{1}-{v}_{2}|$, and

Thus, all the conditions of Theorem 3.3 are satisfied and consequently, its conclusion applies to the problem (3.6).

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The authors declare that they have no competing interests.

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Each of the authors SKN and MO contributed to each part of this study equaly and read and approved the final version of the manuscript.

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Ntouyas, S.K., Obaid, M. A coupled system of fractional differential equations with nonlocal integral boundary conditions.
*Adv Differ Equ* **2012, **130 (2012). https://doi.org/10.1186/1687-1847-2012-130

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### Keywords

- Caputo fractional derivative
- fractional differential systems
- integral boundary conditions
- fixed-point theorems