Advances in Mathematical Physics

Volume 2013 (2013), Article ID 426061, 5 pages

http://dx.doi.org/10.1155/2013/426061

Research Article

Existence of Solutions for Fractional Differential Inclusions with Separated Boundary Conditions in Banach Space

Mabrouk Bragdi,¹ Amar Debbouche,² and Dumitru Baleanu3,4,5

¹Department of Mathematics, Laboratory of Dynamical Systems and Control, Larbi Ben M'hidi University, P.O. Box 358, Oum El Bouaghi, Algeria

²Department of Mathematics, Guelma University, 24000 Guelma, Algeria

³Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey

⁴Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

⁵Institute of Space Sciences, Magurele-Bucharest, Romania

Received 10 March 2013; Accepted 11 May 2013

Academic Editor: Changpin Li

Copyright © 2013 Mabrouk Bragdi 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.

We discuss the existence of solutions for a class of some separated boundary differential inclusions of fractional orders involving the Caputo derivative. In order to obtain necessary conditions for the existence result, we apply the fixed point technique, fractional calculus, and multivalued analysis.

1. Introduction

In recent years a great interest was devoted to the study of (singular and Neumann) boundary-value problems of fractional order [1–8], see also [9–12]. In the literature of fractional calculus, there are several definitions of fractional derivative that can be used. However, the most popular senses are the Riemann-Liouville and Caputo fractional derivatives; see, for instance, [13–19].

In this paper, we use the Caputo's fractional derivative since mathematical modeling of many physical problems requires initial and boundary conditions. These demands are satisfied using the Caputo fractional derivative. For more details we refer the reader to [20, 21] and references therein.

The importance of fractional boundary-value problems stems from the fact that they model various applications in fluid mechanics, viscoelasticity, physics, biology, and economics which cannot be modeled by differential equations with integer derivatives [21–23].

Delbosco and Rodino [24] considered the existence of a solution for the nonlinear fractional differential equation , where and , is a given function, continuous in. They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Qiu and Bai [25] considered the existence of positive solution for equation:

where and with being singular at (i.e., ), by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone. Recently, Aitaliobrahim [26] considered the existence of solutions to the boundary-value problem:

where is a closed multifunction, measurable in the first argument, and Lipschitz continuous in the second argument, by using fixed point theory for multivalued maps.

Motivated by the previous results, in this work we establish the existence result of a new version of fractional separated boundary-value problem:

where is nonconvex, closed multifunction, measurable in the first argument, and Lipschitz continuous in the second argument, and are in a Banach space . The work is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel while in Section 3, we give the main result. Finally in Section 4 we give example to illustrate the application of our results.

2. Preliminaries

In this section, we present basic definitions of fractional calculus and some essential facts from multivalued analysis that will be used in this work to obtain our main results.

Definition 1. A real function , is said to be in the space , if there exists a real number , such that , where , and it is said to be in the space if , .

Definition 2. The left Riemann-Liouville fractional integral of order , of a function , is defined by

Definition 3. For , and , the left Caputo fractional derivative is defined by

where is the well-known Gamma function.

The Caputo derivative defined in (5) is related to the Riemann-Liouville fractional integral, , of order , by

It is known (see [20]) that

where in (7), .

Now, let be a real separable Banach space with the norm . We denote by the Banach space of continuous functions from to equipped with the norm . For and for nonempty sets of we denote and . A multifunction is said to be measurable if its graph is measurable.

Also, we recall the following results that will be used in this paper.

Definition 4 (see [26]). Let be a multifunction with closed values:

(i)is -Lipschitz if for each ,(ii) is a contraction if it is -Lipschitz with ,(iii) has a fixed point if there exists such that .

Lemma 5 (see [26]). If is a contraction with nonempty closed values, then it has a fixed point.

Lemma 6 (see [26]). Assume that is a multifunction with nonempty closed values satisfying the following:

(i)for every is measurable on ;(ii)for every is (Hausdorff) continuous on .

Then for any measurable function , the multifunction is measurable on .

Definition 7 (see [26]). A measurable multivalued function is said to be integrably bounded if there exists a function such that for all for almost every .

Definition 8 (see [26]). A function is said to be a solution of (3) if is absolutely continuous on and satisfies (3).

Lemma 9 (see [25]). Given and , the unique solution of

is

where

Obviously, is continuous on and , for each and some .

3. Main Results

Now we are in a position to state and prove the main results of this paper.

Theorem 10. Let be a set-valued map with nonempty closed values satisfying the following:

(i)for each is measurable and integrably bounded;(ii)there exists a function such that for all and for all

Then, if , for all , the problem (3) has at least one solution on .

Proof. For the proof of this theorem, we use the similar steps as those of [26, Theorem 2.6] together with the theory of fractional calculus. Let be in . We introduce first the function defined by

and the multifunction defined by

for all . Consider the following problem:

We should note that the function is a solution of (15), if and only if the function is a solution of (3), for all .

Next, by Lemma 6, for is closed and measurable; then it has a measurable selection which, by hypothesis (i), belongs to . Thus the set

is nonempty. Let us transform problem (15) into a fixed point problem. Consider the multivalued map

defined as follows, for :

where

We will show that satisfies the assumptions of Lemma 5. The proof will be given in two steps.

Step  1   (T has nonempty closed values). Indeed, let such that converges to in . Then and for each ,

where is the Aumann integral of , which is defined as

Using the fact that the set-valued map is closed and by (14), we conclude that the set

is closed for all . Then

So, there exists such that

Hence . So is closed for each .

Step  2   (T is a contraction). Indeed, let and consider . Then there exists such that

Using (14), there exists such that

On the other hand, let and consider the valued map , given by

We claim that is nonempty, for each . Indeed, let ; then we have

Hence, there exists , such that

The multifunction

Then there exists a measurable selection for denoted by such that, for all ,

and

Now, for all , set and

We have

So, we conclude that

By an analogous relation, obtained by interchanging the roles of and , it follows that

By letting , we obtain

Consequently, if , is a contraction. By Lemma 5, has a fixed point which is a solution of (15).

4. Example

In this section we present an example to illustrate the applications of our main results, and we consider the following fractional inclusion boundary-value problem:

where and are arbitrary real numbers and is a multivalued map given by

Then we have

Let . Then for some .

By Theorem 10 the problem (38) has at least one solution on .

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