Abstract and Applied Analysis

Volume 2014 (2014), Article ID 571536, 10 pages

http://dx.doi.org/10.1155/2014/571536

Research Article

Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations

Xianghu Liu and Yanfang Li

School of Mathematics and Computer Science, Zunyi Normal College, Zunyi 563002, China

Received 1 January 2014; Accepted 22 March 2014; Published 16 June 2014

Academic Editor: Chuanzhi Bai

Copyright © 2014 Xianghu Liu and Yanfang Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper is concerned with the sufficient conditions for the existence of solutions for a class of generalized antiperiodic boundary value problem for nonlinear fractional impulsive differential equations involving the Riemann-Liouville fractional derivative. Firstly, we introduce the fractional calculus and give the generalized R-L fractional integral formula of R-L fractional derivative involving impulsive. Secondly, the sufficient condition for the existence and uniqueness of solutions is presented. Finally, we give some examples to illustrate our main results.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration; it is also as old as ordinary differential calculus. For the last decades, fractional differential equations have been receiving intensive attention because they provide an excellent tool for the description of memory and hereditary properties of various materials and processes, such as physics, mechanics, chemistry, and engineering; for more details, one can see Kilbas et al. [1] and Podlubny [2] and the references therein.

There have been considerable developments in the theory of impulsive differential equations in the last few decades. Impulsive differential equations have become more important in some mathematical models of real phenomena, especially in control, biology, medicine, and information (see [3, 4]). So the study of fractional impulsive differential equations is a more meaningful work. Some significant developments in fractional impulsive differential equations with Caputo derivative have been presented [5–27]. Recently, Fečkan et al. defined the solutions for fractional impulsive differential equations with Caputo derivative (for more details, see [17]). They considered the Cauchy problems for the following impulsive fractional differential equations:

where   , are constants.    denotes Caputo’s fractional derivative. Some sufficient conditions for existence of the solutions have been established by applying Schaefer’s fixed point theorem, Banach fixed point theorem, and the theorem of nonlinear alternative of Leray-Schauder type.

But as far as we know, there are few papers that consider the fractional impulsive differential equations with Riemann-Liouville derivative (only see [15, 24]).

Motivated by [15, 17, 24] and some related literature, we study the existence and uniqueness of solutions for the generalized antiperiodic boundary value problem for fractional differential equations with impulsive effects

where is the Riemann-Liouville fractional derivative, ,  ,   ,   are given constants and

and and denote the left and the right limit of at , respectively.

For clarity and brevity, we restrict that the impulsive functions are constants , . Indeed, we can also define the impulsive functions as .

Remark 1. For , (2) reduces to the first order nonlinear impulsive differential equation with antiperiodic boundary value problem.

To the best of the authors’ knowledge, no one has studied the existence of solutions for (2). The purpose of this paper is to study the existence and uniqueness of solution of the generalized antiperiodic boundary value problem for nonlinear fractional impulsive differential equation involving Riemann-Liouville fractional derivative by using some fixed point theorems.

2. Preliminaries and Lemmas

In this section, we introduce notations, definitions, and preliminaries that will be used in this paper. In order to define the solution of (2), we will consider the following spaces.

,  ; there exist and with ,  .

,  ; there exist with ,  .

, where .

It is easy to check that the space is a Banach space with norm

Let us recall the following known definitions. For more details see [1].

Definition 2. Let be a finite interval on the real axis . The Riemann-Liouville fractional integral of order is defined by

Definition 3. The Riemann-Liouville derivative of order ,   can be written as

Lemma 4 (see Lemma 2.5 in [1]). Let ,   and let be the fractional integral of order . If and , we have the following equality:

Lemma 5. If   ,  , then

where .

Proof. If , then we obtain

Suppose ; the result holds; that is,

When , we obtain that

The proof is completed.

Lemma 6. Let , , . If and , then for , one has

for , , one has

where

Proof. Firstly, according to the fractional integral definitions, we get

If , by Lemma 4, the result is easily to get.

If , integrating by parts repeatedly, we obtain

where the integral

where using the substitution .

So, if , by (15), we have

If , , integrating by parts and using Lemma 5 and (17) repeatedly, we get

By (15), if ,  , we have

The proof is completed.

Remark 7. In Lemma 6, if the assumption is replaced by , we will get the same result of Lemma 4.

Lemma 8. The impulsive antiperiodic boundary value problem

where , has a unique solution given by

where

Proof. Let be a solution of (21). By Lemma 6  , we have

where .

According to the following properties:

we obtain

Then by the antiperiodic boundary value condition, we have

Conversely, assuming that is a solution of the impulsive fractional integral equation (22), we can obtain the impulsive fractional differential equation (21).

This completes the proof.

3. Main Results

This section deals with the existence and uniqueness of solutions for the problem (2).

Firstly, for , , we define an operator as

Theorem 9. If the following condition is satisfied:

 (H1):there exist constants such that

 then the fractional impulsive differential equation (2) has at least one solution.

Proof. Assume (H1) hold; let

and define .

When , , for , by (H1), we have

which implies that .

In view of the continuity of , we get that the operator is continuous easily.

Next, we will prove that is a completely continuous operator.

For , , if , , when , by (H1), we have

According to the Ascoli-Arzela theorem, we can obtain which is a completely continuous operator. Therefore, by Schauder’s fixed point theorem, the operator has at least one fixed point, which implies that fractional impulsive differential equation (2) has at least one solution .

Theorem 10. Assume that

 (H2):there exists constant such that

Then problem (2) has a unique solution if

Proof. We define and choose

Firstly, we prove that , where .

For , by (H2), we have

Next, for , by (H2), we get

According to inequality (34), we obtain that the operator is a contractive mapping on . Hence, by Banach fixed point theorem, problem (2) has a unique solution.

The proof is completed.

4. Examples

Example 1. Choose , , and , and consider the following fractional impulsive generalized antiperiodic boundary value problem:

where

Let ,  ; clearly, assumption (H1) is satisfied. By Theorem 9, the fractional impulsive generalized antiperiodic boundary value problem (38) has at least one solution.

Example 2. Choose , , and ; consider the following fractional impulsive generalized antiperiodic boundary value problem:

where

Letting , condition (H2) of Theorem 10 can be verified, so Example 2 has at least one solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their careful reading of the paper and insightful comments that will improve the quality of the paper. They would also like to acknowledge the valuable comments and suggestions from the editors that will vastly contribute to the improvement of the presentation of the paper. This project is supported by NNSF of China Grant nos. 11271087 and 61263006, Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116, the Open Fund of Guangxi Key Laboratory of Hybrid Computation, and IC Design Analysis no. 2012HCIC07.

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