ISRN Mathematical Analysis

VolumeΒ 2011, Article IDΒ 468346, 14 pages

http://dx.doi.org/10.5402/2011/468346

## Existence Results for a Coupled System of Nonlinear Fractional Differential Equation with Four-Point Boundary Conditions

^{1}Faculty of Education Al-Arish, Suez Canal University, Ismailia 41522, Egypt^{2}Faculty of Computers and Informatics, Suez Canal University, Ismailia 41522, Egypt

Received 30 August 2011; Accepted 20 October 2011

Academic Editor: J.-L.Β Wu

Copyright Β© 2011 M. Gaber and M. G. Brikaa. 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

This paper studies a coupled system of nonlinear fractional differential equation with four-point boundary conditions. Applying the Schauder fixed-point theorem, an existence result is proved for the following system: , , , , , , , , where satisfy certain conditions.

#### 1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymerrheology, and so forth involves derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much importance and attention. For details, see [1β8] and the refrences therein.

On the other hand, the study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature, for instance, see [9, 10]. Recently, in [11], the existence of nontrivial solutions was investigated for a coupled system of nonlinear fractional differential equations with two-point boundary conditions by using Schauder's fixed-point theorem. Reference [12] established the existence of a positive solution to a singular coupled system of fractional order. The existence of nontrivial solutions for a coupled system of nonlinear fractional differential equations with three-point boundary conditions was investigated in [13] by using Schauder's fixed point theorem.

In this paper, we consider a four-point boundary value problem for a coupled system of nonlinear fractional differential equation given by where 1, ,,,, , , , , , , , , , , , is the standard Riemann-Liouville fractional derivative, and are given continuous function.

The organization of this paper is as follows. In Section 2, we present some necessary definition and preliminary results that will be used to prove our main results. The proofs of our main results are given in Section 3.

#### 2. Preliminaries

For the convenience of the reader, we present here the necessary definition from fractional calculus theory and preliminary results.

*Definition 2.1 (see [5]). *The Riemann-Liouville fractional integral of order of function is given by
provided that the integral exists.

*Definition 2.2 (see [5]). *The Riemann-Liouville fractional derivative of order of function is given by
where and denotes the integral part of number , provided that the right side is pointwise defined on .

Lemma 2.3 ([5]). *Let , exists for . Then,
*

*Remark 2.4. *The following properties are useful for our discussion: , , ,β, ; , , and .

For convenience, we introduce the following notation. Let

Let Denote the space of all continuous functions defined on . Let be a Banach space endowed with the norm , where , , see [11] Lemma 3.2, and let be a Banach space equipped with the norm , where , . Thus, is a Banach with the norm defined by for .

Lemma 2.5. *Let be a given function and . Then, the unique solution of
**
is given by
**
where is Green's function given by
*

*Proof. *For , , the general solution of (2.5) can be written as

By the boundary condition,

By the boundary condition,

Substituting (2.11) into (2.12), we get

Substituting (2.13) into (2.11), we get

Thus, the unique solution of (2.5) and (2.6) is
where is given by (2.8).

Similarly, the general solution of is where , can be obtained from by replacing with . Let denote Green's function for the boundary value problem (1.1).

Consider the coupled system of integral equation:

#### 3. Main Results

Lemma 3.1. *Assume that are continuous functions. Then, is a solution of (1.1) if and only if is a solution of (2.18).*

*Proof. *The proof is immediate from Lemma 2.5, so we omit it.

Let us define an operator as where

In view of the continuity of , it follows that is continuous. Moreover, by Lemma 3.1, the fixed-point of the operator coincides with the solution of (1.1).

For the forthcoming analysis, we introduce the growth condition on and asthere exists a nonnegative function such that there exist a nonnegative function such that

Let us set the following notations for convenience

Define a ball in the Banach space as where .

Theorem 3.2. *Assume that the assumptions () and () hold. Then, there exists a solution for the four-point boundary value problem (1.1).*

* Proof. *As a first step, we prove that :

Thus,

Similarly, it can be shown that . Hence, we conclude that .

Since are continuous on , therefore, .

Now, we show that is a completely continuous operator. For that we fix

For , we have

Analogously, it can be proved that

Since the functions are uniformly continuous on , therefore, it follows from the above estimates that is an equicontinuous set. Also, it is uniformly bounded as . Thus, we conclude that is a completely continuous operator. Hence, by Schauder's fixed-point theorem, there exists a solution for the four-point boundary value problem (1.1).

#### References

- A. Babakhani and V. D. Gejji, βExistence of positive solutions of nonlinear fractional differential equations,β
*Journal of Mathematical Analysis and Applications*, vol. 278, no. 2, pp. 434β442, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - Z. Bai and H. Lu, βPositive solutions for boundary value problem of nonlinear fractional differential equation,β
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495β505, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - M. El-Shahed and J. J. Nieto, βNontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,β
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3438β3443, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Jankowski, βPositive solutions to second order four-point boundary value problems for impulsive differential equations,β
*Applied Mathematics and Computation*, vol. 202, no. 2, pp. 550β561, 2008. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - 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, Amsterdam, The Netherlands, 2006. - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. - I. Podlubny, βGeometric and physical interpretation of fractional integration and fractional differentiation,β
*Fractional Calculus & Applied Analysis*, vol. 5, no. 4, pp. 367β386, 2002. View at Google Scholar Β· View at Zentralblatt MATH - S. Zhang, βPositive solution for boundary value problem of nonlinear frctional differential equations,β
*Electronic Journal of Differential Equations*, vol. 2006, no. 36, pp. 1β12, 2006. View at Google Scholar - Y. Chen and H.-L. An, βNumerical solution of coupled Burgers equations with time and space fractional derivatives,β
*Applied Mathematics and Computation*, vol. 200, no. 1, pp. 87β95, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - V. Gafiychuk, B. Datsko, V. Meleshko, and D. Blackmore, βAnalysis of the solutions of coupled nonlinear fractional reaction-diffusion equations,β
*Chaos, Solitons and Fractals*, vol. 41, no. 3, pp. 1095β1104, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - X. Su, βBoundary value problem for a coupled system of nonlinear fractional differential equations,β
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 64β69, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - C. Z. Bai and J. X. Fang, βThe existence of a positive solution for a singular coupled system of nonlinear fractional differential equations,β
*Applied Mathematics and Computation*, vol. 150, no. 3, pp. 611β621, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - B. Ahmad and J. J. Nieto, βExistence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,β
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838β1843, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet