1. Introduction

The discretization of the ordinary differential equations is an important and necessary step towards finding their numerical solutions. Instead of the standard discretization based on the arithmetic progression, one can use an equally efficient -discretization related to geometric progression. This alternative method leads to -difference equations, which in the limit correspond to the classical differential equations. -difference equations are found to be quite useful in the theory of quantum groups [1]. For historical notes and development of the subject, we refer the reader to [2, 3] while some recent results on -difference equations can be found in [4–6]. However, the theory of boundary value problems for nonlinear -difference equations is still in the initial stages, and many aspects of this theory need to be explored.

Differential inclusions arise in the mathematical modelling of certain problems in economics, optimal control, stochastic analysis, and so forth and are widely studied by many authors; see [7–13] and the references therein. For some works concerning difference inclusions and dynamic inclusions on time scales, we refer the reader to the papers [14–17].

In this paper, we study the existence of solutions for second-order -difference inclusions with nonseparated boundary conditions given by where is a compact valued multivalued map, is the family of all subsets of , is a fixed constant, and is a fixed real number.

The aim of our paper is to establish some existence results for the Problems (1.1)-(1.2), when the right-hand side is convex as well as nonconvex valued. First of all, an integral operator is found by applying the tools of -difference calculus, which plays a pivotal role to convert the given boundary value problem to a fixed-point problem. Our approach is simpler as it does not involve the typical series solution form of -difference equations. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we will combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we will use the fixed-point theorem for generalized contraction multivalued maps due to Wegrzyk. The methods used are standard; however, their exposition in the framework of Problems (1.1)-(1.2) is new.

The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel, and we prove our main results in Section 3.

2. Preliminaries

In this section, we introduce notation, definitions, and preliminary facts which we need for the forthcoming analysis.

2.1. -Calculus

Let us recall some basic concepts of -calculus [1–3].

For , we define the -derivative of a real-valued function as The higher-order -derivatives are given by The -integral of a function defined in the interval is given by and for , we denote provided the series converges. If and is defined in the interval , then Similarly, we have Observe that and if is continuous at , then In -calculus, the integration by parts formula is

2.2. Multivalued Analysis

Let us recall some basic definitions on multivalued maps [18, 19].

Let denote a normed space with the norm . A multivalued map is convex (closed) valued if is convex (closed) for all . is bounded on bounded sets if is bounded in for all bounded sets in (i.e., . is called upper semicontinuous (u.s.c.) on if for each , the set is a nonempty closed subset of , and if for each open set of containing , there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every bounded set in . If the multivalued map is completely continuous with nonempty compact values, then is u.s.c. if and only if has a closed graph (i.e., , , imply ). has a fixed-point if there is such that . The fixed-point set of the multivalued operator will be denoted by .

For more details on multivalued maps, see the books of Aubin and Cellina [20], Aubin and Frankowska [21], Deimling [18], and Hu and Papageorgiou [19].

Let denote the Banach space of all continuous functions from into with the norm Let be the Banach space of measurable functions which are Lebesgue integrable and normed by

Definition 2.1. A multivalued map with nonempty compact convex values is said to be measurable if for any , the function is measurable.

Definition 2.2. A multivalued map is said to be Carathéodory if(i) is measurable for each ,(ii) is upper semicontinuous for almost all .Further a Carathéodory function is called -Carathéodory if(iii)for each , there exists such that for all and for a.e. .

Let be a Banach space, let be a nonempty closed subset of , and let be a multivalued operator with nonempty closed values. is lower semicontinuous (l.s.c.) if the set is open for any open set in . Let be 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 . A subset of is decomposable if for all and measurable, the function , where stands for the characteristic function of .

Definition 2.3. If is a multivalued map with compact values and , then is of lower semicontinuous type if is lower semicontinuous with closed and decomposable values.

Let be a metric space associated with the norm . The Pompeiu-Hausdorff distance of the closed subsets is defined by where .

Definition 2.4. A function is said to be a strict comparison function (see [25]) if it is continuous strictly increasing and , for each .

Definition 2.5. A multivalued operator on with nonempty values in is called(a)-Lipschitz if and only if there exists such that (b)a contraction if and only if it is -Lipschitz with ,(c)a generalized contraction if and only if there is a strict comparison function such that

The following lemmas will be used in the sequel.

Lemma 2.6 (see [22]). Let be a Banach space. Let be an -Carathéodory multivalued map with , and let be a linear continuous mapping from to , then the operator defined by has compact convex values and has a closed graph operator in .

In passing, we remark that if , then for any with as in Lemma 2.6.

Lemma 2.7 (nonlinear alternative for Kakutani maps [23]). 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 ,(ii)or there is a and with .

Lemma 2.8 (see [24]). Let be a separable metric space, and let be a lower semicontinuous multivalued map with closed decomposable values, then has a continuous selection; that is, there exists a continuous mapping (single-valued) such that for every .

Lemma 2.9 (Wegrzyk's fixed-point theorem [25, 26]). Let be a complete metric space. If is a generalized contraction with nonempty closed values, then .

Lemma 2.10 (Covitz and Nadler's fixed-point theorem [27]). Let be a complete metric space. If is a multivalued contraction with nonempty closed values, then has a fixed-point such that , that is, .

3. Main Results

In this section, we are concerned with the existence of solutions for the Problems (1.1)-(1.2) when the right-hand side has convex as well as nonconvex values. Initially, we assume that is a compact and convex valued multivalued map.

To define the solution for the Problems (1.1)-(1.2), we need the following result.

Lemma 3.1. Suppose that is continuous, then the following problem has a unique solution where is the Green's function given by

Proof. In view of (2.7) and (2.9), the solution of can be written as where , are arbitrary constants. Using the boundary conditions (1.2) and (3.4), we find that Substituting the values of and in (3.4), we obtain (3.2).

Let us denote

Definition 3.2. A function is said to be a solution of (1.1)-(1.2) if there exists a function with a.e. and where is given by (3.3).

Theorem 3.3. Suppose that (H1)the map has nonempty compact convex values and is Carathéodory,(H2)there exist a continuous nondecreasing function and a function such that for each ,(H3)there exists a number such that

then the BVP (1.1)-(1.2) has at least one solution.

Proof. In view of Definition 3.2, the existence of solutions to (1.1)-(1.2) is equivalent to the existence of solutions to the integral inclusion Let us introduce the operator
We will show that satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.
Step 1 ( is convex for each ). Indeed, if , belong to , then there exist such that for each , we have Let , then, for each , we have Since is convex (because has convex values); therefore, Step 2 ( maps bounded sets into bounded sets in ). Let be a bounded set in and , then for each , there exists such that Then, in view of (H2), we have Thus, Step 3 ( maps bounded sets into equicontinuous sets of ). Let , and be a bounded set of as in Step 2 and . For each   The right-hand side tends to zero as . As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli Theorem, we can conclude that is completely continuous.Step 4 ( has a closed graph). Let , and . We need to show that . means that there exists such that, for each , We must show that there exists such that, for each , Clearly, we have Consider the continuous linear operator defined by From Lemma 2.6, it follows that is a closed graph operator. Moreover, we have Since , it follows from Lemma 2.6 that for some .Step 5 (a priori bounds on solutions). Let be a possible solution of the Problems (1.1)-(1.2), then there exists with such that, for each , For each , it follows by (H2) and (H3) that Consequently, Then by (H3), there exists such that .
Let The operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by Lemma 2.7, it follows that has a fixed-point in which is a solution of the Problems (1.1)-(1.2). This completes the proof.

Next, we study the case where is not necessarily convex valued. Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.

Theorem 3.4. Suppose that the conditions (H2) and (H3) hold. Furthermore, it is assumed that (H4) has nonempty compact values and(a) is measurable,(b) is lower semicontinuous for a.e. ,(H5)for each , there exists such that then, the BVP (1.1)-(1.2) has at least one solution.

Proof. Note that (H4) and (H5) imply that is of lower semicontinuous type. Thus, by Lemma 2.8, there exists a continuous function such that for all . So we consider the problem Clearly, if is a solution of (3.31), then is a solution to the Problems  (1.1)-(1.2). Transform the Problem (3.31) into a fixed-point theorem where We can easily show that is continuous and completely continuous. The remainder of the proof is similar to that of Theorem 3.3.

Now, we prove the existence of solutions for the Problems  (1.1)-(1.2) with a nonconvex valued right-hand side by applying Lemma 2.9 due to Wegrzyk.

Theorem 3.5. Suppose that (H6) has nonempty compact values and is measurable for each ,(H7) for almost all and with and for almost all , where is strictly increasing,then the BVP (1.1)-(1.2) has at least one solution on if is a strict comparison function, where .

Proof. Suppose that is a strict comparison function. Observe that by the assumptions (H6) and (H7), is measurable and has a measurable selection (see Theorem  3.6 [28]). Also and Thus, the set is nonempty for each .
As before, we transform the Problems (1.1)-(1.2) into a fixed-point problem by using the multivalued map given by (3.11) and show that the map satisfies the assumptions of Lemma 2.9. To show that the map is closed for each , let such that in , then and there exists such that, for each , As has compact values, we pass onto a subsequence to obtain that converges to in . Thus, and for each , So, and hence is closed.
Next, we show that Let and . Then, there exists such that for each , From (H7), it follows that So, there exists such that Define as Since the multivalued operator is measurable (see Proposition  3.4 in [28]), there exists a function which is a measurable selection for . So, , and for each , For each , let us define then Thus, By an analogous argument, interchanging the roles of and , we obtain for each . So, is a generalized contraction, and thus, by Lemma 2.9, has a fixed-point which is a solution to (1.1)-(1.2). This completes the proof.

Remark 3.6. We notice that Theorem 3.5 holds for several values of the function , for example, , where , , and so forth. Here, we emphasize that the condition (H7) reduces to for , where a contraction principle for multivalued map due to Covitz and Nadler [27] (Lemma 2.10) is applicable under the condition . Thus, our result dealing with a nonconvex valued right-hand side of (1.1) is more general, and the previous results for nonconvex valued right-hand side of the inclusions based on Covitz and Nadler's fixed-point result (e.g., see [8]) can be extended to generalized contraction case.

Remark 3.7. Our results correspond to the ones for second-order -difference inclusions with antiperiodic boundary conditions (, ) for . The results for an initial value problem of second-order -difference inclusions follow for . These results are new in the present configuration.

Remark 3.8. In the limit , the obtained results take the form of their “continuous” (i.e., differential) counterparts presented in Sections  4 (ii) for of [29].

Example 3.9. Consider a boundary value problem of second-order -difference inclusions given by where and is a multivalued map given by For , we have Thus, with , . Further, using the condition we find that , where . Clearly, all the conditions of Theorem 3.3 are satisfied. So, the conclusion of Theorem 3.3 applies to the Problem (3.47).