We are concerned with the nonlinear

Fractional differential equations (FDEs) have received increasing interest for the last three decades. It is benefited by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in the modeling of many phenomena in various fields of science, engineering economics, and other fields, see for instance [

Babakhani and Gejji [

Stojanović [

On the other hand, realistic problems arising from economics, optimal control, and so on can be modeled as differential inclusions. Recently, El-Sayed and Ibrahim [

Very recently, in the survey paper [

Ouahab [

Chang and Nieto [

However, to the best of our knowledge, the existence of solutions for fractional integro-differential inclusions with multipoint boundary conditions has not been paid much attention. Our goal is to fill this gap in literature.

In the present work, we consider more general fractional integro-differential inclusions

The organization of this paper is as follows. In Section

In this section, we recall some basic definitions and notations and give several lemmas which are useful in our discussion.

The Riemann-Liouville fractional integral of order

The Riemann-Liouville fractional derivative of function

Let

If the fractional derivative

The reader is referred to [

Let

In view of Lemma

Conversely, if

Now, we recall some facts from multivalued analysis.

Let

Consider

A multivalued map

A multivalued map

More details on multivalued maps can be found in the books of Deimling [

The multivalued map

for each

For any

Let

The following Bohnenblust-Karlin fixed-point lemma and the fixed-point theorem for contractive multivalued operators given by Covitz and Nadler are of great importance in the proofs of our main results. The proofs of these results can be found in Bohnenblust and Karlin [

Let

Let

For convenience, let us list some conditions.

Assume that hypothesis (H1) is satisfied. For any

We define the operator

A function

In this section, we present our main results and prove them. Firstly, under convexity condition on the multivalued right-hand side, we are to establish the existence theorem of solutions for fractional differential inclusions (

Assume that hypotheses (H1) and (H2) are satisfied. Then BVP (

where

To transform the problem into a fixed-point problem, we consider the multivalued operator,

For any

We shall show that

we claim that there exists a

In fact, if it is not true, then for any

On the other hand, from (H2), we obtain

Dividing both sides by

This is a contraction to (H3). Hence there exists a

Let

The right-hand side of the above inequality tends to zeros independently of

Let

We need to show that there exists

Therefore,

As a direct corollary of Theorem

Suppose that (H1) and (H2) are satisfied. Then the problem (

there exist functions

We only need to verify that (H4) implies (H3) in Theorem

In the next part, we are concerned with the BVP (

Assume that the following hypotheses hold:

There exist three functions

Transform the problem into a fixed-point problem. Let

Indeed, let

For every

It is clear that

There exists

Let

Consider

Since the multivalued operator

For each

Then, for

So,

By an analogous relation, obtained by interchanging the roles of

In (

This work is supported by the National Natural Science Foundation of PR China (no. 10701023, No. 10971221) and Shanghai Natural Science Foundation (no. 10ZR1400100).