We discuss the existence of solutions for a boundary value problem of Riemann-Liouville fractional differential inclusions of order 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.

1. 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 [13]. 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 fractional-order models more realistic and practical than the integer-order models and has accounted for the popularity of the subject. For some recent development on the topic, see [417] 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 [1823] and the references cited therein.

In this paper, we study the following boundary value problem: where is the standard Riemann-Liouville fractional derivative of order ,is a multivalued map, is the family of all subsets of , and is a continuous function.

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.

2. Preliminaries

2.1. Fractional Calculus

Let us recall some basic definitions of fractional calculus [1, 2].

Definition 1. The Riemann-Liouville derivative of fractional order is defined as 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 3. A function is called a solution of problem (1) if there exists a function with , a.e. such that , a.e. and , .

Lemma 4 (see [17]). Given , then the unique solution of the problem is given by where

Lemma 5 (see [17]). The functions have the following properties: (i), for all ;(ii) for all ;(iii) for all .

2.2. Basic Concepts of Multivalued Maps

Let be a normed space and let be a multivalued map. is said to be(i) convex (closed) valued if is convex (closed) for all ;(ii) bounded on bounded sets if is bounded in for all , where , (i.e., );(iii) upper semicontinuous (u.s.c.) on if the set is a nonempty closed subset of for each and if for each open set of containing there exists an open neighborhood of such that ; (iv) completely continuous if is relatively compact for every .

Definition 6. 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 .

Definition 7. has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .

Definition 8. Let . A multivalued map is said to be measurable if the function is measurable for every .

In the sequel, by we mean a Banach space of continuous functions from into with the norm whereas is the Banach space of measurable functions which are Lebesgue integrable and normed by .

Definition 9. A multivalued map is said to be Carathéodory if the map is measurable for each and the map is upper semicontinuous for almost all . Further, a Carathéodory function is called -Carathéodory if for each , there exists such that for all and for a. e. .

Definition 10. For each , the set of selections of is defined by

Definition 11. For a nonempty closed subset of a Banach space , let be a nonempty multivalued operator with closed values. We call to be lower semi-continuous (l.s.c.) if the set is open for any open set in .

Definition 12. A subset of is measurable if belongs to the -algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in .

Definition 13. A subset of is said be decomposable if for all and measurable , the function , where stands for the characteristic function of .

Definition 14. A multivalued operator has a property (BC) if is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values, where is a separable metric space.

Definition 15. Let be a multivalued map with nonempty compact values. We say that is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator is lower semi-continuous and has nonempty closed and decomposable values, where

Definition 16. Let be a metric space induced from the normed space and let be defined by where and . Then is a metric space and is a generalized metric space (see [24]).

Definition 17. A multivalued operator is called -Lipschitz if and only if there exists such that and a contraction if and only if it is -Lipschitz with .

For further details on multi-valued maps, we refer the reader to the books [25, 26].

3. 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.

Lemma 18 (see [27]). Let be a Banach space. Let be an -Carathéodory multivalued map with compact and convex values and let be a linear continuous mapping from to . Then the operator is a closed graph operator in .

Lemma 19 (nonlinear alternative for Kakutani maps [28]). Let be a Banach space, a closed convex subset of an open subset of , and . Suppose that is an upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of . Then either (i) has a fixed point in , or (ii)there is a and with .

Theorem 20. Assume that is Carathéodory and has nonempty compact and convex values; there exists a continuous nondecreasing function and a function such that there exists a constant such that Then the boundary value problem (1) has at least one solution on .

Proof. In view of Lemma 4, we define an operator by and show that it satisfies the hypotheses of Lemma 19. Since is convex ( has convex values), therefore, it can be easily shown that is convex for each .
As a next step, we prove that maps the bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then, for each , , there exists such that Then, for , , we have Thus, Now we show that maps the bounded sets into equicontinuous sets of .
Let , with and . For each , we obtain Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, therefore, by the Ascoli-Arzelá theorem, it follows that is completely continuous.
In our next step, we show that has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that for each , Thus, it suffices to show that there exists such that for each , Define a linear operator as Notice that Thus, it follows from Lemma 18 that is a closed graph operator. Further, we have , since . Thus, for some , we have In the last step, we show that there exists an open set with for any and all . Let and . Then there exists with such that for , we have and using the computations used in the second step, we obtain In consequence, we have By the assumption , there exists such that . Let us set Observe that the operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by Lemma 19, we have that has a fixed point which is a solution of the problem (1). This completes the proof.

Example 21. Consider the following boundary value problem: where is a multivalued map given by For , we have Here , , , with , and . Using the given values in the condition : 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.

Lemma 22 (see [29, 30]). Let be a separable metric space and let be a multivalued operator satisfying the property (BC). Then has a continuous selection; that is, there exists a continuous function (single-valued) such that for every .

Theorem 23. Suppose that and hold. In addition, we assume the following condition: is a nonempty compact-valued multivalued map such that (a) is measurable, (b) is lower semicontinuous for each . Then the problem (1) has at least one solution on .

Proof. Observe that the assumptions and imply that is of l.s.c. type. Then, by Lemma 22, there exists a continuous function such that for all .
Let us consider the problem One can note that if 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 as It is easy to show that the operator 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].

Lemma 24 ([31]). Let be a complete metric space. If is a contraction, then .

Theorem 25. Assume that is such that is measurable for each ; for almost all and with and for almost all . Then the boundary value problem (1) has at least one solution on if

Proof. By the assumption , it follows that the set is nonempty for each . So has a measurable selection (see [32, Theorem III.6]). Now it will be shown that the operator defined by (17) satisfies the hypotheses of Lemma 24. To show that for each , let be such that in . Then and there exists such that, for each , As has compact values, we pass onto a subsequence (if necessary) to obtain that converges to in . Thus, and for each , we have Hence, .
Next, we show that there exists such that Let and . Then there exists such that, for each , By , we have that So, there exists such that Define by Since the multivalued operator is measurable ([32, Proposition III.4]), there exists a function which is a measurable selection for . So and for each , we have that .
For each , let us define Thus, Hence, Analogously, interchanging the roles of and , we obtain Since is a contraction, it follows from Lemma 24 that has a fixed point which is a solution of (1). This completes the proof.


The authors thank the editor and the reviewer for their constructive comments that led to the improvement of the paper. The research of H. H. Alsulami and B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.