- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

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

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

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

^{1}Department of Mathematics, Amirkabir University of Technology, Hafez Avenue, P.O. Box 15914, Tehran 15875-4413, Iran^{2}Department 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.

#### References

- R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific Publishing, Singapore, 2000. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B. V., Amsterdam, The Netherlands, 2006. - V. Lakshmikantham, S. Leela, and J. V. Devi,
*Theory of Fractional Dynamic Systems*, Cambridge Scientific Publishers, Cambridge, UK, 2009. - B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,”
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 150, no. 3, pp. 611–621, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Chen, D. Chen, and Z. Lv, “The existence results for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions,”
*Bulletin of the Iranian Mathematical Society*, vol. 38, no. 3, pp. 607–624, 2012. View at Google Scholar · View at MathSciNet · View at Scopus - M. Gaber and M. G. Brikaa, “Existence results for a coupled system of nonlinear fractional differential equation with four-point boundary conditions,”
*ISRN Mathematical Analysis*, vol. 2011, Article ID 468346, 14 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - N. Khodabakhshi and S. M. Vaezpour, “Existence results for a coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 73, 15 pages, 2013. View at Google Scholar · View at MathSciNet - X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”
*Applied Mathematics Letters. An International Journal of Rapid Publication*, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - G. Wang, B. Ahmad, and L. Zhang, “A coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on an unbounded domain,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 248709, 11 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J. Wang, H. Xiang, and Z. Liu, “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations,”
*International Journal of Differential Equations*, vol. 2010, Article ID 186928, 12 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - W. Yang, “Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions,”
*Computers & Mathematics with Applications*, vol. 63, no. 1, pp. 288–297, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. Yuan, “Two positive solutions for ($n$-1, 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 2, pp. 930–942, 2012. View at Publisher · View at Google Scholar · View at Scopus - A. Anguraj, P. Karthikeyan, and G. N'Guereata, “Nonlocal Cauchy problem for some fractional abstract integro-differential equations in Banach spaces,”
*Communications in Mathematical Analysis*, vol. 6, no. 1, pp. 31–35, 2009. View at Google Scholar · View at MathSciNet - B. Ahmad and S. Sivasundaram, “Some existence results for fractional integro-differential equations with nonlinear conditions,”
*Communications in Applied Analysis*, vol. 12, no. 2, pp. 107–112, 2008. View at Google Scholar · View at MathSciNet · View at Scopus - K. Balachandran and J. J. Trujillo, “The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 72, no. 12, pp. 4587–4593, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Nazari and S. Shahmorad, “Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 883–891, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Wang, X. Yan, X. H. Zhang, T. M. Wang, and X. Z. Li, “A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions,”
*Opuscula Mathematica*, vol. 31, no. 1, pp. 119–135, 2011. View at Publisher · View at Google Scholar - Y. Liu, “Existence and unboundedness of positive solutions for singular boundary value problems on half-line,”
*Applied Mathematics and Computation*, vol. 144, no. 2-3, pp. 543–556, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Liang and J. Zhang, “Existence of three positive solutions of $m$-point boundary value problems for some nonlinear fractional differential equations on an infinite interval,”
*Computers & Mathematics with Applications*, vol. 61, no. 11, pp. 3343–3354, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus