^{1, 2}

^{1}

^{2}

^{1}

^{2}

This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results.

This paper deals with the existence of solutions for the following fractional-order differential inclusions with impulsive nonlocal conditions:

The nonlocal problem was more general and has better effect than the classical Cauchy problems. So it has been studied extensively under various conditions in the literature [

The theory of impulsive differential equations and differential inclusions has received much attention for the past decades because of its wide applicability in control, electrical engineering, mechanics, biology, and so on. For more details on this theory and applications, we refer to the monograph of Lakshmikantham et al. [

Since fractional-order differential equations have proved to be valuable tools in the modeling of many phenomena in physics and technical sciences, differential equations involving Riemann-Liouville as well as Caputo derivatives have been investigated extensively in the last decades (see [

However, there is little information in the literature on neutral fractional-order impulsive differential inclusions with nonlocal conditions. Motivated by works mentioned above, we consider the existence results of (

This paper is organized as follows. In Section

In this section, we introduce some definitions, notes, and preliminary facts which will be used in this paper.

Let

Let

A mapping

A mapping

A multivalued map

for each

A multivalued operator

a contraction if it is

The key tool in our approach is the following fixed point theorem.

Let

the operator inclusion

there exists an

Let

For more details on multivalued map, see the books of Aubin and Cellina [

Now, we recall some definitions and facts about fractional derivatives and fractional integrals of arbitrary order, see [

The Riemann-Liouville fractional integral operator of order

The Caputo fractional-order derivative of order

Let

Let

In this section, main results are presented.

Let

As a consequence of Lemmas

The function

For the study of the system (

there exist a function

The existence of solutions is now presented.

Assume that (H1)–(H6) are satisfied; then the problem (

We transform the problem (

Consider the multivalued operators

Define an open ball

Indeed, let

In fact, if

By elementary computation, we have

We will prove that there exists

Consider the continuous operator

From Lemma

As a consequence of Lemma

If we take

Assume that (H1)–(H6) are satisfied; then the problem (

From (

Since the proof of Corollary

Assume that (H1), (H3)–(H6), and the following condition are satisfied,

(H2^{'}) There exists a function

Then the problem (

In this case, we take

At last we would like to discuss the impulsive condition.

Assume that (H1)–(H4),

In this section, an example is given to show the effectiveness of our theoretical result. Consider the following fractional-order differential inclusions with impulsive and nonlocal conditions:

Thus,

By computation, we have

Suppose that

Clearly, for suitable

The authors would like to thank the Associate Editor and all the anonymous reviewers for their valuable comments and constructive suggestions, which lead to the improvement of the presentation of this paper. This work was funded by Scientific Research Foundation of Nanjing Institute of Technology under Grant Nos. QKJB2010028, QKJA2011009 and also jointly supported by the National Natural Science Foundation of China under Grant Nos. 60874088 and 11072059 and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20110092110017, the JSPS Innovation Program under Grant CXZZ12_0080.