An Existence Theorem for Fractional -Difference Inclusions with Nonlocal Substrip Type Boundary Conditions

Abstract

By employing a nonlinear alternative for contractive maps, we investigate the existence of solutions for a boundary value problem of fractional -difference inclusions with nonlocal substrip type boundary conditions. The main result is illustrated with the aid of an example.

1. Introduction

In this paper, we consider the following boundary value problem of fractional -difference inclusions with nonlocal and substrip type boundary conditions: where denotes the Caputo fractional -derivative of order , is a multivalued map, is the family of all nonempty subsets of , is a given continuous function, and is a real constant. Here, we emphasize that the nonlocal conditions are regarded as more plausible than the standard initial conditions for the description of some physical phenomena. In (1), may be understood as , where , , are given constants and . For more details, we refer to the work by Byszewski and Lakshmikantham [1, 2].

Boundary value problems with integral boundary conditions constitute an important class of problems and arise in the mathematical modeling of various phenomena such as heat conduction, wave propagation, gravitation, chemical engineering, underground water flow, thermoelasticity, and plasma physics. They include two-point, three-point, multipoint, and nonlocal boundary value problems.

The topic of fractional differential equations has been of great interest for many researchers in view of its theoretical development and widespread applications in various fields of science and engineering such as control, porous media, electromagnetic, and other fields [3, 4]. For some recent results and applications, we refer the reader to a series of papers ([5–13]) and the references cited therein.

Fractional -difference (-fractional) equations are regarded as fractional analogue of -difference equations. Motivated by recent interest in the study of fractional-order differential equations, the topic of -fractional equations has attracted the attention of many researchers. The details of some recent work on the topic can be found in ([14–20]). For notions and basic concepts of -fractional calculus, we refer to a recent text [21].

The present work is motivated by a recent paper of the authors [22], where the problem (1) was considered for a single valued case. To the best of our knowledge, this is the first paper dealing with fractional -difference inclusions in the given framework. Moreover, the main result of our paper can be regarded as an improvement and extension of some known results; see, for instance, [18, 19].

The paper is organized as follows. Section 2 contains some fundamental concepts of fractional -calculus. In Section 3, we show the existence of solutions for the problem (1) by means of the nonlinear alternative for contractive mappings. Finally, an example illustrating the applicability of our result is presented.

2. Preliminaries

First of all, we recall the notations and terminology for -fractional calculus [21, 23, 24].

For a real parameter , a -real number denoted by is defined by

The -analogue of the Pochhammer symbol (-shifted factorial) is defined as The -analogue of the exponent is The -gamma function is defined as where . Observe that .

Definition 1 (see [23]). Let be a function defined on . The fractional -integral of the Riemann-Liouville type of order is and

Observe that in Definition 1 yields -integral: For more details on -integrals and fractional -integrals, see Sections 1.3 and 4.2, respectively, in [21].

Remark 2. The -fractional integration possesses the semigroup property ([21, Proposition 4.3]): Further, it has been shown in Lemma 6 of [24] that

Before giving the definition of fractional -derivative, we recall the concept of -derivative.

We know that the -derivative of a function is defined as Furthermore,

Definition 3 (see [21]). The Caputo fractional -derivative of order is defined by where is the smallest integer greater than or equal to .

Next, we recall some properties involving Riemann-Liouville -fractional integral and Caputo fractional -derivative ([21, Theorem 5.2]):

Lemma 4 (see [22]). Let . Then, the following problem is equivalent to an integral equation: where

3. Existence Results

First of all, we outline some basic definitions and results for multivalued maps [25, 26].

For a normed space , let is closed}, is bounded}, is compact}, and is compact and convex}. A multivalued map (i)is convex (closed) valued if is convex (closed) for all ;(ii)is bounded on bounded sets if is bounded in for all (i.e., );(iii)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 ;(iv)is said to be completely continuous if is relatively compact for every ;(v)is said to be measurable if, for every , the function is measurable;(vi)has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .

For each , define the set of selections of by

Definition 5. A multivalued map is said to be a Carathéodory function 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. .

We define the graph of to be the set and recall two results for closed graphs and upper-semicontinuity.

Lemma 6 (see [25, Proposition 1.2]). If is u.s.c., then is a closed subset of , that is, for every sequence and , if , , , and , then . Conversely, if is completely continuous and has a closed graph, then it is upper semicontinuous.

Lemma 7 (see [27]). Let be a separable Banach space. Let be an -Carathéodory function. Then, the operator is a closed graph operator in .

Definition 8. A function is called a solution of problem (1) if there exists a function with , a.e. on such that , a.e. on and and .

To prove our main result in this section we will use the following form of the nonlinear alternative for contractive maps [28, Corollary 3.8].

Theorem 9. Let be a Banach space and let be a bounded neighborhood of . Let and be two multivalued operators such that (a) is contraction,(b) is u.s.c and compact. Then, if , either(i) has a fixed point in or(ii)there is a point and with .

Theorem 10. Assume that (H₁) is -Carathéodory multivalued map;(H₂) there exists a continuous nondecreasing function and a function such that (H₃) is a continuous function satisfying the condition (H₄) there exists a number such that  with , where Then, the problem (1) has at least one solution on .

Proof. To transform the problem (1) into a fixed point problem, we define an operator as Next, we introduce two operators and as follows: Observe that . We will show that the operators and satisfy all the conditions of Theorem 9 on . For the sake of clarity, we split the proof into a number of steps and claims.
Step 1. is a contraction on . This is a consequence of (H₃). Indeed, we have Taking supremum over , we have
Step 2. is compact, convex valued, and completely continuous. This will be established in several claims.
Claim 1. maps bounded sets into bounded sets in . For that, let be a bounded set in . Then, for each , , we have and consequently, for each , we have
Claim 2. maps bounded sets into equicontinuous sets. As before, let be a bounded set and let for . Let with and . Then, for each , we obtain which is independent of and tends to zero as . Therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
Claim 3. has a closed graph. Let , , and . Then, we need to show that . Associated with , there exists such that, for each , Then, we have to show that there exists such that, for each ,
Let us consider the continuous linear operator defined by
Observe that as . Thus, it follows by Lemma 7 that is a closed graph operator. Further, we have . Since , therefore, we have for some . Hence, has a closed graph (and therefore has closed values). In consequence, the operator is compact valued.
Thus, the operators and satisfy hypotheses of Theorem 9 and hence, by its application, it follows that either condition (i) or condition (ii) holds. We show that the conclusion (ii) is not possible. If for , then there exists such that , that is, In consequence, we have If condition (ii) of Theorem 9 holds, then there exist and with , where . Then, is a solution of with . Now, by the last inequality, we get which contradicts (23). Hence, has a fixed point on by Theorem 9, and consequently the problem (1) has a solution. This completes the proof.

Example 11. Consider the following -fractional boundary value problem:

Here, , , , , , , and Obviously, With the given values, it is found that , , , , , and by (H₃). Thus, all the conditions of Theorem 10 hold. In consequence, the conclusion of Theorem 10 applies to the boundary value problem (40).

Disclosure

Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.