In this paper, a new existence result is obtained for a fractional multivalued problem with fractional integral boundary conditions by applying a (Krasnoselskii type) fixed-point result for multivalued maps due to Petryshyn and Fitzpatric [Trans. Am. Math. Soc. 194:1-25, 1974]. The case for lower semi-continuous multivalued maps is also discussed. An example for the illustration of our main result is presented.
MSC: 34A60, 34A08.
Keywords:fractional differential inclusions; nonlocal boundary conditions; fixed-point theorems
The theory of fractional differential equations and inclusions has developed into an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [1-4]. The nonlocal behavior exhibited by a fractional-order differential operator makes it distinct from the integer-order differential operator. It means that the future state of a dynamical system or process involving fractional derivatives depends on its current state as well its past states. In fact, differential equations of arbitrary order are capable of describing memory and hereditary properties of several materials and processes. This characteristic of fractional calculus has contributed to its popularity and has convinced many researchers of the need to shift their focus from classical integer-order models to fractional-order models. There has been a great surge in developing new theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see [5-18] and the references cited therein.
In this paper, we consider the following boundary value problem of fractional differential inclusions with fractional integral boundary conditions:
We establish two new existence results for the problem (1.1). The first result relies on a nonlinear alternative for contractive maps, while in the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semi-continuous multivalued maps with nonempty closed and decomposable values.
The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel and Section 3 deals with the main results.
provided the integral exists.
To define the solutions of the problem (1.1), we need the following lemma. Though the proof of this lemma involves standard arguments, we trace its proof for the convenience of the reader.
Proof It is well known  that the solution of fractional differential equation in (2.2) can be written as
Substituting these values in (2.3) yields
we find that
To establish the main results of this paper, we use the following form of the nonlinear alternative for contractive maps [, Corollary 3.8].
Lemma 2.6 (Lasota and Opial )
3 Existence results
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
In the sequel, we set
Theorem 3.2Assume that
Combining the inequality (3.5) with the corresponding one obtained by interchanging the roles of x and y, we get
Step 2. We shall show that the operator is u.s.c. and compact. It is well known [, Proposition 1.2] that if an operator is completely continuous and has a closed graph, then it is u.s.c. Therefore we will prove that is completely continuous and has a closed graph. This step involves several claims.
which implies that
Therefore the operators and satisfy all the conditions of Theorem 2.4. So the conclusion of Theorem 2.4 applies and either condition (i) or condition (ii) holds. We show that the conclusion (ii) is not possible. If for , then there exist and such that
3.1 The lower semi-continuous case
This section is devoted to the study of the case that the maps in (1.1) are not necessarily convex-valued. We establish the existence result for the problem at hand by applying the nonlinear alternative of Leray-Schauder type and a selection theorem due to Bressan and Colombo  for lower semi-continuous maps with decomposable values. Before presenting this result, we revisit some basic concepts.
Let be a nonempty closed subset of a Banach space and be a multivalued operator with nonempty closed values. is lower semi-continuous (l.s.c.) if the set is open for any open set in . Let be a subset of . is measurable if belongs to the σ algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in ℝ. A subset of is decomposable if for all and measurable , the function , where stands for the characteristic function of .
which is called the Nemytskii operator associated with F.
Definition 3.4 Let be a multivalued function with nonempty compact values. We say F is of lower semi-continuous type (l.s.c. type) if its associated Nemytskii operator ℱ is lower semi-continuous and has nonempty closed and decomposable values.
Lemma 3.5 ()
LetYbe a separable metric space and letbe a multivalued operator satisfying the property (BC). Thenhas a continuous selection, that is, there exists a continuous function (single-valued) such thatfor every.
Theorem 3.6Assume that (H2), (H4), (H5), and the following condition hold:
Consider the problem
Clearly are continuous. Also the argument in Theorem 3.2 guarantees that and satisfy all the conditions of the nonlinear alternative for contractive maps in the single-valued setting  and hence the problem (3.8) has a solution. □
Example 3.7 Consider the following fractional boundary value problem:
The authors declare that they have no competing interests.
Each of the authors, AA, SKN, BA and HHA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant no. 3-130/1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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