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We investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion

In this paper, we will consider the existence of solutions for the following multipoint boundary value problem of a fractional order differential inclusion

The present paper is motivated by a recent paper of Liang and Zhang [

Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, and engineering. For details, see [

The existence of solutions of initial value problems for fractional order differential equations has been studied in the literature [

The aim here is to establish existence results for problem (

In Section

In this section, we present some notations and preliminary lemmas that will be used in the proof of the main result.

Let

Recall that the Pompeiu-Hausdorff distance of the closed subsets

Also, we denote by

Let

We define the graph of

If

For the convenience of the reader, we present here the following nonlinear alternative of the Leray-Schauder type and its consequences.

Let

there is a

The multifunction

If

Let

Consider the following.

A set-valued map

A set-valued map

Finally, the following results are easily deduced from the theoretical limit set properties.

Let

Let

Let

it is integrably bounded; that is, there exists

the image sequence

The following important result follows from the Dunford-Pettis theorem (see [

Every semicompact sequence

When the nonlinearity takes convex values, Mazur’s Lemma, 1933, may be useful.

Let

Let

the functions from

the functions from

The Riemann-Liouville fractional integral operator of order

The Riemann-Liouville fractional derivative of order

The equality

Let

Let

By

From [

In what follows,

For any

Note that

The function

By calculation, it is easy to prove that Lemma

Now we are able to present the existence results for problem (

To obtain the complete continuity of existence solutions operator, the following lemma is still needed.

Let

The Carathéodory multivalued map

There exists a continuous nondecreasing function

There exists a constant

Then, problem (

Let

We show that

On the other hand, we get

Therefore, with Lemma

Condition (H1) implies that

In view of (H2), there exists

Note that the operator

Now we prove the existence of solutions for the problem (

A multivalued operator

a contraction if and only if it is

Let

A measurable multivalued function

Assume that the following condition holds:

There exist

Then, the boundary value problem (

We transform problem (

Note that since the set-valued map

We will prove that

First, we note that since

Second, we prove that

Next we show that

Consider the fractional boundary value problem,

Also, by direct calculation, we can obtain that

The authors thank the referees for their careful reading of this paper and useful suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.