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

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

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][6][7][8][9][10][11][12][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][15][16][17][18][19][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 2 The Scientific World Journal means of the nonlinear alternative for contractive mappings. Finally, an example illustrating the applicability of our result is presented.
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 > 0 is defined by where ⌈ ⌉ is the smallest integer greater than or equal to .

Existence Results
First of all, we outline some basic definitions and results for multivalued maps [25,26]. (iv) is said to be completely continuous if (B) is relatively compact for every B ∈ P ( ); (v) is said to be measurable if, for every ∈ R, 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 Fix .
We define the graph of to be the set ( ) = {( , ) ∈ × , ∈ ( )} and recall two results for closed graphs and upper-semicontinuity.
Lemma 7 (see [27]). Let be a separable Banach space. Let 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].
with ℓ 0 < 1, where Then, the problem (1) Observe that F = F 1 + F 2 . We will show that the operators F 1 and F 2 satisfy all the conditions of Theorem 9 on [0, 1].
For the sake of clarity, we split the proof into a number of steps and claims.
Step 1. F 1 is a contraction on ([0, 1], R). This is a consequence of (H 3 ). Indeed, we have Taking supremum over ∈ [0, 1], we have Step 2. F 2 is compact, convex valued, and completely continuous. This will be established in several claims.
and consequently, for each ℎ ∈ F 2 ( ), we have Claim 2. F 2 maps bounded sets into equicontinuous sets. As before, let be a bounded set and let ℎ ∈ F 2 ( ) for ∈ .

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The Scientific World Journal Thus, the operators F 1 and F 2 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 ∈ F 1 ( ) + F 2 ( ) for ∈ (0, 1), then there exists ∈ , such that = F( ), that is, In consequence, we have If condition (ii) of Theorem 9 holds, then there exist ∈ (0, 1) and ∈ with = F( ), where = { ∈ ([0, 1], R) : ‖ ‖ ≤ }. Then, is a solution of = F( ) with ‖ ‖ = . Now, by the last inequality, we get which contradicts (23). Hence, F has a fixed point on [0, 1] by Theorem 9, and consequently the problem (1) has a solution. This completes the proof.