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Mabrouk Bragdi, Amar Debbouche, Dumitru Baleanu, "Existence of Solutions for Fractional Differential Inclusions with Separated Boundary Conditions in Banach Space", Advances in Mathematical Physics, vol. 2013, Article ID 426061, 5 pages, 2013. https://doi.org/10.1155/2013/426061
Existence of Solutions for Fractional Differential Inclusions with Separated Boundary Conditions in Banach Space
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.
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  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  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  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.
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.
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 ). 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 ). If is a contraction with nonempty closed values, then it has a fixed point.
Lemma 6 (see ). 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 ). A measurable multivalued function is said to be integrably bounded if there exists a function such that for all for almost every .
Lemma 9 (see ). 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).
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 .
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