Abstract

In this paper, we investigate the existence of solutions for a class of fractional boundary value problems with anti-periodic boundary value conditions with -Caupto fractional derivative. By means of some standard fixed point theorems, sufficient conditions for the existence of solutions for the fractional differential inclusions with -Caputo derivatives are presented. Our result generalizes the known special case if and single known results to the multi-valued ones.

1. Introduction

Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary noninteger order [1, 2], which is a wonderful technique to understand of memory and hereditary properties of materials and processes. Some recent contributions to fractional differential equations have been carried out, see the monographs [3–6], and the references cited therein. Much attention has been focused on the study of anti-periodic boundary conditions, which are applied in different fields, such as blood flow problems, chemical engineering, underground water flow, populations dynamics, and so on, see the references ([7–9]) and paper cited therein. In 2009, Ahamad and Otero-Espinar [7] investigated the following fractional inclusions with anti-periodic boundary conditions

where is the standard Caputo derivative of order is a multivalued map, is the family of all subsets of . Some sufficient conditions for the existence of solutions are given by means of Bohnenblust-Karlin fixed point theorem.

There are several definitions of fractional differential derivatives and integrals, such like Caputo type, Rimann–Liuville type, Hadamard type, and Erdelyi-Kober type and so on. In order to develop the fractional calculus, special kernels and some form of differential operator are chosen, see [10–16]. The -Caputo fractional derivative of order , was first introduced by Almeida in [4]. Some properties, like semigroup law, Taylor’s Theorem, Fermat’s Thorem, etc., were presented. This new defined fractional derivative could model more accurately the process using differential kernels for the fractional operator.

In 2018, Samet and Aydi in [17] considered the following fractional differential boundary value problem with anti-periodic boundary conditions:

where is the -Caputo fractional derivative of order and is a given function. A Lyapunov-type inequality is established. The authors also give some examples to illustrate the applications of their main results.

Inspired by the above works, we investigate the following anti-periodic fractional inclusions with -Caputo derivatives:

where . is the -Caputo fractional derivative of order and is a multivalued map, is the family of all subsets of . Sufficient conditions for the existence of solutions are given in view of the fixed point theorems for multi-valued mapping. The exposition in the framework of problem is new. If taking , , , the fractional differential inclusions (3) reduces to the fractional differential inclusions (1). If we take , where is a given continuous function, then the problem (3) corresponds to the single-valued problem (2). The rest of this paper is organized as follows. We first present some basic definitions of fractional calculus, -Caputo derivative and multi-valued maps. In Section 3, the main results on the existence of solutions for integral boundary value problem (3) are presented. An example is given to illustrate our main result in the last section.

2. Preliminaries

In this section, we recall some notations, definitions and preliminaries about fractional calculus [18–20], and -Caputo fractional calculus [4, 17, 21–23].

Definition 1 [9]. The Caputo fractional integral order of a function is given bythat is,Let be a given function such that

Definition 2 [18]. The factional integral of order of a function with respect to is defined by

Definition 3 [4]. The -Caputo fractional derivative of order of a function is defined as

Remark 1. Similarly, for and , , the definition of -Caputo fractional derivative of order of a function could be given as follows:

The following are definitions and properties concerning multi-valued maps [9, 24, 25] which will be used in the remainder. A multivalued map :(i) Is called upper semicontinuous(u.s.c.) on , if for each the set is a nonempty closed subset of , and for each open set of containing , there exists an open neighborhood of such that .(ii) The graph of is defined by the set .(iii) is said to be measurable if for every , the function

Let be a separable metric space and let be a multivalued operator. We call has a property (BC) if is lower semi-continuous (l.s.c.) and has nonempty closed and decomposable values. Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with as

for a.e. which is called the Nemyskii operator associated with . Let be a multivalued function with nonempty compact values. We say 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. Let be a subset of . A is measurable if belongs to the -algebra generated by all sets of the form , where is the 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 . If the multi-valued map is completely continuous with nonempty compact values , then T is U.s.c. if and only if has a closed graph. For each , is a closed interval from to , denote the selection set of as

Let . The Pompeiu-Hausdorff distance of is defined by

where .

For convenience, we present the following notations.

To set the frame for our main results, we introduce the following lemmas.

Lemma 1 [26]. Let be a complete metric space. If is a contraction, then

Lemma 2 [27]. (Nonlinear alternative for Kakutani maps). Let be a Banach space, a closed convex subset of , an open subset of and . Suppose that is a 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 such that .

Lemma 3 [28]. Let be a Banach space, and be a -Carathédory set-valued map with and let be a linear continuous mapping. Then the set-valued map defined byis a closed graph operator in

Lemma 4 [24]. Let be a separable metric space and 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 .

Lemma 5 [29]. Let , Then is a solution toif and only ifwhereThat is,

Lemma 6 [17]. If the problemscould be transformed into the following problemswhere A nontrivial solution to (22) is given byi.e.,From Lemma 6, we can easily know that

3. Main Results

Now we are in the position to state our main results.

3.1. The Lipschitz Case

is such that, for every is measurable.

There exists for almost all such that

with for almost all

Theorem 1. Suppose that . Ifthen problem (3) has at least a solution in .

Proof. By Lemma 6, we define the operator as follows:We shall prove that the operator satisfies all the conditions in Lemma 1, thus has a fixed point that is a solution to the antiperiodic problem (3). First of all, for each the operator is closed. Let be such that in . Then , and there exists such that for each As has compact values, we pass onto a subsequence to get that converges to Thus, and for each we haveThus, .
Next, we will show there exists such thatIn fact, let and There exists such that for each From , we obtainThus, there exists such thatDefine bySince the multivalued operator is measurable, there exists a function which is a measurable selection for So and for each we haveFor each defineand one hasHence, we haveThe same arguments discussed as (40), interchanging and yieldsBy (28), is a contraction. Thus, by Lemma 1, we conclude that admits a fixed point which is a solution to problem (3). It completes the proof.