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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 , denote the Caputo fractional derivatives, , , and .
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 endowed with the norm . Obviously, is a Banach space. Also, let endowed with the norm . The product space is also a Banach space with norm .
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 , the Caputo derivative of fractional order q is defined as
where 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 , . Then and , for all .
Lemma 2.4 Let , . Then , for all .
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 . Then for a given , the solution of the fractional differential equation
subject to the boundary condition
is given by
Proof For some constant , 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 in (2.4), we obtain (2.3). □
3 Main results
For the sake of convenience, we set


and
Define the operator by

The first result is based on Leray-Schauder alternative.
Lemma 3.1 (Leray-Schauder alternative, [23] p.4)
Let be a completely continuous operator (i.e., a map that restricted to any bounded set in E is compact). Let
Then either the set is unbounded, or F has at least one fixed point.
Theorem 3.2 Suppose that and . Assume that there exist real constants , () and , such that (), we have

In addition, it is assumed that
where and 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 is completely continuous. By continuity of functions f and g, the operator T is continuous.
Let be bounded. Then there exist positive constants and such that
Then for any , 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 . Then we have

Analogously, we can obtain

Therefore, the operator is equicontinuous, and thus the operator is completely continuous.
Finally, it will be verified that the set is bounded. Let , then . For any , we have
Then
and
Hence, we have
and
which imply that
Consequently,
for any , where is defined by (3.3), which proves that 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 are continuous functions and there exist constants , , such that for all and , ,
and
In addition, assume that
where and are given by (3.1) and (3.2), respectively. Then the boundary value problem (1.1) has a unique solution.
Proof Define and such that
We show that , where .
For , we have

Hence,
In the same way, we can obtain that
Consequently, .
Now for , and for any , we get

and consequently we obtain
Similarly,
It follows from (3.4) and (3.5) that
Since , 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, , , , , , , , , and and . Note that and . Furthermore, , , 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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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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DOI: https://doi.org/10.1186/1687-1847-2012-130
Keywords
- Caputo fractional derivative
- fractional differential systems
- integral boundary conditions
- fixed-point theorems