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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 149659, 8 pages
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.
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.
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, [1–4]. 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 [5–9]. Recently, in , 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
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.
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 .
For computational convenience, we introduce the following constant:
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 ). 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 .
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 .
Lemma 7 (nonlinear alternative for single valued maps, see ). 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,
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.
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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