Existence of Solutions for Fractional Differential Inclusions with Separated Boundary Conditions in Banach Space

1 Department of Mathematics, Laboratory of Dynamical Systems and Control, Larbi Ben M’hidi University, P.O. Box 358, Oum El Bouaghi, Algeria 2 Department of Mathematics, Guelma University, 24000 Guelma, Algeria 3 Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey 4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 5 Institute of Space Sciences, Magurele-Bucharest, Romania

In this paper, we use the Caputo's fractional derivative since mathematical modeling of many physical problems requires initial and boundary conditions.These demands are satisfied using the Caputo fractional derivative.For more details we refer the reader to [20,21] and references therein.
The importance of fractional boundary-value problems stems from the fact that they model various applications in fluid mechanics, viscoelasticity, physics, biology, and economics which cannot be modeled by differential equations with integer derivatives [21][22][23].
The work is organized as follows.In Section 2, we recall some preliminary facts that we need in the sequel while in Section 3, we give the main result.Finally in Section 4 we give example to illustrate the application of our results.

Preliminaries
In this section, we present basic definitions of fractional calculus and some essential facts from multivalued analysis that will be used in this work to obtain our main results.
Also, we recall the following results that will be used in this paper.
Lemma 5 (see [26]).If  :  → 2  is a contraction with nonempty closed values, then it has a fixed point.

Main Results
Now we are in a position to state and prove the main results of this paper.Proof.For the proof of this theorem, we use the similar steps as those of [26, Theorem 2.6] together with the theory of fractional calculus.Let ,  be in .We introduce first the function  : [0, 1] →  defined by and the multifunction  : [0, 1]×C([0, 1], ) → 2  defined by