Existence Results for a Riemann-Liouville-Type Fractional Multivalued Problem with Integral Boundary Conditions

We discuss the existence of solutions for a boundary value problem of Riemann-Liouville fractional differential inclusions of order α ∈ (2, 3] with integral boundary conditions. We establish our results by applying the standard tools of fixed point theory for multivalued maps when the right-hand side of the inclusion has convex as well as nonconvex values. An illustrative example is also presented.


Introduction
In the last few decades, fractional calculus is found to be an effective modeling tool in many branches of physics, economics, and technical sciences [1][2][3].A fractional-order differential operator is nonlocal in its character in contrast to its counterpart in classical calculus.It means that the future state of a dynamical system or process based on fractional-order derivative depends on both its current and past states.Thus, the application of fractional calculus in various materials and processes enables an investigator to study the complete behavior (ranging from past to current states) of such stuff.This is indeed an important feature that makes fractionalorder models more realistic and practical than the integerorder models and has accounted for the popularity of the subject.For some recent development on the topic, see [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein.
Differential inclusions appear in the mathematical modeling of certain problems in economics, optimal control, and so forth and are widely studied by many authors.Examples and details can be found in a series of papers [18][19][20][21][22][23] and the references cited therein.
Here we remark that the present work is motivated by a recent paper [17], where problem (1) is considered with  as single valued and the results on existence and nonexistence of positive solutions are obtained.
The main tools of our study include nonlinear alternative of Leray-Schauder type, a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps, and Covitz and Nadler's fixed point theorem for contraction multivalued maps.The application of these results is new in the framework of the problem at hand.We recall some preliminaries in Section 2 while the main results are presented in Section 3.

Fractional Calculus.
Let us recall some basic definitions of fractional calculus [1,2].
provided the integral exists, where [] denotes the integer part of the real number .
Definition 2. The Riemann-Liouville fractional integral of order  is defined as provided the integral exists.
Definition 17.A multivalued operator H : X → P  (X) is called -Lipschitz if and only if there exists  > 0 such that and a contraction if and only if it is -Lipschitz with  < 1.
For further details on multi-valued maps, we refer the reader to the books [25,26].

Existence Results for the Multivalued Problem
In this section, we present some existence results for the problem (1).Our first result deals with the case when  is Carathéodory.We make use of the following known results to establish the proof.
Proof.In view of Lemma 4, we define an operator and show that it satisfies the hypotheses of Lemma 19.Since  , is convex ( has convex values), therefore, it can be easily shown that Q is convex for each  ∈ ([0, 1], R).
In our next step, we show that Q has a closed graph.Let   →  * , ℎ  ∈ Q(  ), and ℎ  → ℎ * .Then we need to show that ℎ * ∈ Q( * ).Associated with ℎ  ∈ Q(  ), there exists Thus, it suffices to show that there exists V * ∈  , * such that for each  ∈ [0, 1], Define a linear operator  : Notice that Thus, it follows from Lemma 18 that  ∘   is a closed graph operator.Further, we have ℎ  () ∈ ( ,  ), since   →  * .Thus, for some V * ∈  , * , we have In the last step, we show that there exists an open set V ⊆ ([0, 1], R) with  ∉ Q() for any  ∈ (0, 1) and all  ∈ V.Let  ∈ (0, 1) and  ∈ Q().Then there exists V ∈  1 ([0, 1], R) with V ∈  , such that for  ∈ [0, 1], we have and using the computations used in the second step, we obtain In consequence, we have By the assumption (H 3 ), there exists M such that ‖‖ ̸ = M. Let us set Observe that the operator Q : V → P(([0, 1], R)) is upper semicontinuous and completely continuous.From the choice of V, there is no  ∈ V such that  ∈ Q() for some  ∈ (0, 1).Consequently, by Lemma 19, we have that Q has a fixed point  ∈ V which is a solution of the problem (1).This completes the proof.
Example 21.Consider the following boundary value problem: where  : [0, 1] × R → P(R) is a multivalued map given by For  ∈ , we have Here  = ∫ Using the given values in the condition (H 3 ): we find that Clearly, all the conditions of Theorem 20 are satisfied.Hence, the conclusion of Theorem 20 applies to the problem (31).
In our next result, we assume that  is not necessarily convex valued.We complete the proof of this result by applying the nonlinear alternative of Leray-Schauder type together with the selection theorem of Bressan and Colombo [29] for lower semi-continuous maps with decomposable values, which is stated below.
Theorem 23.Suppose that (H 2 ) and (H 3 ) hold.In addition, we assume the following condition: One can note that if  ∈  2 ([0, 1], R) is a solution of (36), then  is a solution to the problem (1).To convert the problem (36) to a fixed point problem, we define an operator Q as It is easy to show that the operator Q is continuous and completely continuous.The rest of the proof is similar to that of Theorem 20.So we omit it.This completes the proof.
Finally we show the existence of solutions for the problem (1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued maps according to Covitz and Nadler [31].
Theorem 25.Assume that Proof.By the assumption (H 5 ), it follows that the set  , is nonempty for each  ∈ ([0, 1], R).So  has a measurable selection (see [32,Theorem III.6]).Now it will be shown that the operator Q defined by ( 17) satisfies the hypotheses of Lemma 24.To show that Q As  has compact values, we pass onto a subsequence (if necessary) to obtain that V  converges to V in  1 ([0, 1], R).Thus, V ∈  , and for each  ∈ [0, 1], we have Hence,  ∈ Q.
If the multivalued map  is completely continuous with nonempty compact values, then  is u.s.c.if and only if  has a closed graph; that is,   ∈ (  ) imply that  * ∈ ( * ) when   →  * ,   →  * .