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We discuss the existence of solutions for a boundary value problem of Riemann-Liouville fractional differential inclusions of order

In the last few decades, fractional calculus is found to be an effective modeling tool in many branches of physics, economics, and technical sciences [

Differential inclusions appear in the mathematical modeling of certain problems in economics, optimal control, and so forth and are widely studied by many authors. Examples and details can be found in a series of papers [

In this paper, we study the following boundary value problem:

Here we remark that the present work is motivated by a recent paper [

The main tools of our study include nonlinear alternative of Leray-Schauder type, a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps, and Covitz and Nadler's fixed point theorem for contraction multivalued maps. The application of these results is new in the framework of the problem at hand. We recall some preliminaries in Section

Let us recall some basic definitions of fractional calculus [

The Riemann-Liouville derivative of fractional order

The Riemann-Liouville fractional integral of order

A function

Given

The functions

Let

convex (closed) valued if

bounded on bounded sets if

upper semicontinuous (u.s.c.) on

completely continuous if

If the multivalued map

Let

In the sequel, by

A multivalued map

For each

For a nonempty closed subset

A subset

A subset

A multivalued operator

Let

Let

A multivalued operator

For further details on multi-valued maps, we refer the reader to the books [

In this section, we present some existence results for the problem (

Let

Let

there is a

Assume that

there exists a continuous nondecreasing function

there exists a constant

In view of Lemma

As a next step, we prove that

Let

In our next step, we show that

Consider the following boundary value problem:

In our next result, we assume that

Let

Suppose that

Observe that the assumptions

Let us consider the problem

Finally we show the existence of solutions for the problem (

Let

Assume that

By the assumption

Next, we show that there exists

For each

The authors thank the editor and the reviewer for their constructive comments that led to the improvement of the paper. The research of H. H. Alsulami and B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.