Abstract and Applied Analysis

Volume 2014 (2014), Article ID 297314, 7 pages

http://dx.doi.org/10.1155/2014/297314

Research Article

Solvability for a Coupled System of Fractional Integrodifferential Equations with -Point Boundary Conditions on the Half-Line

Payam Nasertayoob,¹ S. Mansour Vaezpour,¹ and Dumitru Baleanu²

¹Department of Mathematics, Amirkabir University of Technology, Hafez Avenue, P.O. Box 15914, Tehran 15875-4413, Iran

²Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey

Received 20 May 2014; Accepted 1 July 2014; Published 23 July 2014

Academic Editor: Guo-Cheng Wu

Copyright © 2014 Payam Nasertayoob et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to study the solvability for a coupled system of fractional integrodifferential equations with multipoint fractional boundary value problems on the half-line. An example is given to demonstrate the validity of our assumptions.

1. Introduction

The theory of derivatives and integrals of fractional order has undergone rapid development over the years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, mechanics, engineering, and so on [1–3]. Recently, the existence of solutions for coupled systems involving fractional differential equations is one of the theoretical fields investigated by many authors [4–13].

Very recently, Wang et al. [10] studied the existence of solutions for the following coupled system of nonlinear fractional differential equations by using Schauder’s fixed point theorem:

where , , and and denote Riemann-Liouville fractional derivatives of order and order , respectively; also , are such that and .

Motivated by [10], in this paper, we consider a coupled system of nonlinear fractional integrodifferential equations on an unbounded domain and more general boundary conditions:

where , , , are real numbers, , and denotes Riemann-Liouville fractional derivative. It is clear that boundary value problem (2) includes problem (1) as special case.

Integrodifferential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. In particular, some physical phenomena involving certain type of memory effects are represented by integrodifferential equations [14–18].

However, to the best of our knowledge, no work has been reported on the existence results for coupled system of nonlinear fractional integrodifferential equations on an unbounded domain.

The paper is organized as follows. In Section 2, we recall some basic definitions, notations, and preliminary facts. Section 3 is devoted to the existence results for system of nonlinear fractional integrodifferential equations on an unbounded domain. In Section 4, an example is given to demonstrate the applicability of our results.

2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which are used in what follows and can be found in [2, 19].

Definition 1. The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side is pointwise defined.

Definition 2. The Riemann-Liouville fractional derivative of order of a continuous function is given by

where , provided that the right-hand side is pointwise defined.

Remark 3. The following properties are well known:

Lemma 4. For , the equation is valid if and only if

where is the smallest integer greater than or equal to .

Lemma 5. Assume that ; then,

where is the smallest integer greater than or equal to .

For any we can define the space

equipped with the norm

Clearly, is a Banach space [19]. For we define

then, is a Banach space.

3. Main Results

In this section, we prove the existence results for the boundary value problem (2). For convenience we use the following notation:

By replacing with , respectively, we can define .

Lemma 6. Let and ; then, the unique solution of

is given by

where is Green’s function given by

with

Proof. By Lemma 5, the solution of (12) can be writen as

Using the boundary conditions (13), we find that and

Now considering the second boundary condition, we have

Therefore, the unique solution of the boundary value problem (12)–(14) is

where , , and are defined by (16), (17), and (18), respectively. The proof is complete.

Now, we introduce the following function:

where

Remark 7. From the definition of and , for any , we have

where

Let an operator be defined by

From the definition of operator , the problem (2) has a solution if and only if the operator has a fixed point.

Theorem 8. Assume the following.

(H ₁)There exist nonnegative functions such that

where is the beta-function.(H ₂)There exist nonnegative functions such that

where is the beta-function.

Then, the system (2) has a solution.

Proof. Take

and define a ball

At the first step, we prove that the operator transforms the ball into itself. For any we have

In a similar way, we can get

Hence, and this shows that .

Next, we show that is completely continuous. First, Let as in . From (32) we have

Then, by the Lebsegue dominated convergence theorem and continuity of , we obtain

as . Therefore, by Remark 7, we have

as . Similar process can be repeated for ; thus, operator is continuous.

Now, we show that is equicontinuous operator. Let and ; without loss of generality, we may assume that . Since and , for any and , we have

In view of (37), by the similar process used in [20], we can easily prove that operator is equicontinuous. Similar process can be repeated for ; thus, is equicontinuous. On the other hand, is uniformly bounded as . Therefore, is completely continuous operator. Hence, by Schauder fixed point theorem the boundary value problem (2) has at least one solution in .

4. An Example

Consider the following boundary value problem on unbounded domain:

Here , , , , , , , , , , , , and . We have

For

by direct calculation we obtain , , , and

Thus all the conditions of Theorem 8 are satisfied and the problem (38) has at least one solution.

5. Conclusion

In the current paper, we have studied the existence results for a coupled system of nonlinear fractional integrodifferential equations with -point fractional boundary conditions on an unbounded domain. The result obtained in this paper is based on Schauder’s fixed point theorem. In order to show the validity of the assumptions made in our result, we also include an illustrative example.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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