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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 149659, 8 pages
http://dx.doi.org/10.1155/2013/149659
Research Article

Boundary Value Problems for a Class of Sequential Integrodifferential Equations of Fractional Order

1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received 16 January 2013; Accepted 13 March 2013

Academic Editor: Jose Luis Sanchez

Copyright © 2013 Bashir Ahmad and Juan J. Nieto. 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

We investigate the existence of solutions for a sequential integrodifferential equation of fractional order with some boundary conditions. The existence results are established by means of some standard tools of fixed point theory. An illustrative example is also presented.

1. Introduction

Nonlinear boundary value problems of fractional differential equations have received a considerable attention in the last few decades. One can easily find a variety of results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional equations in the literature on the topic. The interest in the subject has been mainly due to the extensive applications of fractional calculus in the mathematical modeling of several real-world phenomena occurring in physical and technical sciences; see, for example, [14]. An important feature of a fractional order differential operator, distinguishing it from an integer-order differential operator, is that it is nonlocal in nature. It means that the future state of a dynamical system or process based on a fractional operator depends on its current state as well as its past states. Thus, differential equations of arbitrary order are capable of describing memory and hereditary properties of some important and useful materials and processes. This feature has fascinated many researchers, and they have shifted their focus to fractional order models from the classical integer-order models. For some recent work on the topic, we refer, for instance, to [59]. Recently, in [10], the authors studied sequential fractional differential equations with three-point boundary conditions.

In this paper, we consider a nonlinear Dirichlet boundary value problem of sequential fractional integrodifferential equations given by where denotes the Caputo fractional derivative of order , denotes Riemann-Liouville integral with ,   are given continuous functions, , and are real constants. We also study the fractional integro-differential equation (1) subject to the following boundary conditions:

2. Linear Fractional Differential Equations

For , we consider the following linear fractional differential equation: where denotes the Caputo fractional derivative of order . Rewriting (1) as , we can write its solution as where are arbitrary constants. Now, (6) can be expressed as Differentiating (7), we obtain which can alternatively be written as

Integrating from to , we have where and are arbitrary constants, and

Lemma 1. The unique solution of the linear equation (5) subject to the Dirichlet boundary conditions (2) is given by

Proof. Observe that the general solution of (5) is given by (10). Using the given boundary conditions in (10), we find that Substituting the values of and in (10) yields the solution (12). This completes the proof.

In the next two lemmas, we present the unique solutions of (5) with different kinds of boundary conditions. We do not provide the proofs for these lemmas as they are similar to that of Lemma 1.

Lemma 2. The unique solution of the problem (5)–(3) is given by

Lemma 3. The unique solution of (5) with the boundary conditions (4) is

3. Existence Results for the Nonlinear Problems

Let denote the Banach space of all continuous functions from into endowed with the usual norm defined by .

In view of Lemma 1, we transform problem (1)-(2) to an equivalent fixed point problem as where is defined by

In a similar manner, we can define a fixed point operator for the nonlinear problem (1)–(3) as follows: A fixed point operator for the nonlinear problem (1)–(4) is defined by

We only present the existence results for the problem (1)-(2). Observe that problem (1)-(2) has solutions if the operator equation (16) has fixed points.

For computational convenience, we introduce the following constant:

Theorem 4. Assume that are continuous functions satisfying the following condition: Then, the boundary value problem (1)-(2) has a unique solution if , where and is given by (20).

Proof. Let us define , where are finite numbers given by . Selecting , we show that , where . For , we have which means that .
Now, for , we obtain By the given assumption, ,   is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).

Our next existence result relies on Krasnoselskii's fixed point theorem.

Lemma 5 (Krasnoselskii, see [11]). Let be a closed, convex, bounded, and nonempty subset of a Banach space . Let be the operators such that (i) whenever , (ii) is compact, and continuous, and (iii) is a contraction mapping. Then, there exists such that .

Theorem 6. Let be continuous functions satisfying assumption , and Then, the problem (1)-(2) has at least one solution on provided that where .

Proof. Let us fix and consider . We define the operators and on as For , we find that
Thus, . It follows from assumption together with (25) that is a contraction mapping. Continuities of and imply that the operator is continuous. Also, is uniformly bounded on   as Now, we prove the compactness of the operator . In view of , we define Consequently, we have which is independent of and tends to zero as . Thus, is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Lemma 5 are satisfied. So, by the conclusion of Lemma 5, problem (1)-(2) has at least one solution on .

Now, we show the existence of solutions for the problem (1)-(2) via Leray-Schauder alternative.

Lemma 7 (nonlinear alternative for single valued maps, see [12]). Let be a Banach space, a closed, convex subset of an open subset of , and . Suppose that is a continuous, compact (that is, is a relatively compact subset of ) map. Then, either(i)has a fixed point in , or (ii)there is a (the boundary of in ) and with .

Theorem 8. Let be continuous functions and the following assumptions hold.There exist functions , and nondecreasing functions such that ,  for  all  . There exists a constant such that Then, the boundary value problem (1)-(2) has at least one solution on .

Proof. Consider the operator with , where We show that maps bounded sets into bounded sets in . For a positive number , let be a bounded set in . Then, Consequently,
Next, we show that maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then, we obtain Obviously, the right hand side of the previous inequality tends to zero independently of as . As satisfies the previous assumptions, therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
The proof will be complete by the application of the Leray-Schauder nonlinear alternative (Lemma 7) once we establish the boundedness of the set of all solutions to equations for .
Let be a solution. Then, for , and using the computations in proving that is bounded, we have Consequently, we have In view of , there exists such that . Let us set Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 7), we deduce that has a fixed point which is a solution of the problem (1)-(2). This completes the proof.

Example 9. Consider a boundary value problem of integro-differential equations of fractional order given by where ,  ,  ,  ,  ,  ,  . With the given data, it is found that as ,  , and Clearly, and . Thus, all the assumptions of Theorem 4 are satisfied. Hence, by the conclusion of Theorem 4, the problem (40) has a unique solution.

Acknowledgment

The authors thank the anonymous referees for their valuable comments. The research of J. J. Nieto has been partially supported by Ministerio de Economia y Competitividad (Spain), project MTM2010-15314, and co-financed by the European Community fund FEDER.

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