Discrete Dynamics in Nature and Society

Volume 2013, Article ID 308024, 12 pages

http://dx.doi.org/10.1155/2013/308024

## Generalized Antiperiodic Boundary Value Problems for the Fractional Differential Equation with p-Laplacian Operator

^{1}Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, China^{2}Department of Mathematics and Physics, Anyang Institute of Technology, Anyang, Henan 455000, China

Received 3 February 2013; Accepted 12 March 2013

Academic Editor: Hua Su

Copyright © 2013 Zhi-Wei Lv and Xu-Dong Zheng. 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

We discuss the existence of solutions about generalized antiperiodic boundary value problems for the fractional differential equation with p-Laplacian operator , , , , , , where is the Caputo fractional derivative, , , , and , , , . Our results are based on fixed point theorem and contraction mapping principle. Furthermore, three examples are also given to illustrate the results.

#### 1. Introduction

Fractional differential equations arise in various areas of science and engineering, such as physics, mechanics, chemistry, and engineering. The fractional order models become more realistic and practical than the classical integer models. Due to their applications, fractional differential equations have gained considerable attentions; one can see [1–14] and references therein.

Anti-periodic boundary value problems occur in the mathematical modeling of a variety of physical processes. Anti-periodic problems constitute an important class of boundary value problems and have received considerable attention (see [15–19]).

In [20], Zhang considered the existence and multiplicity results of positive solutions for the following boundary value problem of fractional differential equation: where is a real number, is the Caputo fractional derivative, and is continuous.

In [15], the authors discussed some existence results for the following anti-periodic boundary value problem for fractional differential equations: where is the Caputo fractional derivative of order ; is a given continuous function.

In [16], the authors investigated the following anti-periodic boundary value problem for higher-order fractional differential equations: where is the Caputo fractional derivative of order ; is a given continuous function.

In [17], the authors investigated a class of anti-periodic boundary value problem of fractional differential equations where is the Caputo fractional derivative of order ; is a given continuous function.

In this paper, we discuss the existence of solutions about generalized anti-periodic boundary value problems for the fractional differential equation with p-Laplacian operator where is the Caputo fractional derivative, , , , , and , , , .

If we take , and , then the problem (5) becomes the problem studied in [17]. In this paper, we let .

This paper is organized as follows. In Section 2, we present some background materials and preliminaries. Section 3 deals with some existence results. In Section 4, three examples are given to illustrate the results.

#### 2. Background Materials and Preliminaries

*Definition 1 (see [21]). * The fractional integral of order with the lower limit for a function is defined as
where is the gamma function.

*Definition 2 (see [21]). * Caputo's derivative of order with the lower limit for a function can be written as

Lemma 3 (see [22]). *Assume that with a fractional derivative of order that belongs to . Then
**
where , , .*

Lemma 4. *Let . Then the fractional differential equation
**
has a unique solution which is given by
*

*Proof. * From Lemma 3, we have
Thus,
By , we have
Using the boundary condition and (13), we obtain
Thus,

#### 3. Main Results

Let denote the Banach space of continuous functions and from endowed with the norm defined by where Define an operator as From (18), we conclude that Then (5) has a solution if and only if the operator has a fixed point.

Theorem 5. * Let be continuous. Assume that meets the following condition: there exist , such that
**
Then the problem (5) has at least one solution on for
*

*Proof. * From , we know that is continuous.

Let
For , we have
This, together with (21) and (22), yields that
Hence, is uniformly bounded.

Next we show that is equicontinuous.

For any , , we have
Thus, we conclude that is equicontinuous on , and
By Schauder fixed point theorem we know that there exists a solution for the boundary value problem (5).

Theorem 6. * Let be continuous. Assume that meets the following condition: there exist , such that
**
Then the problem (5) has a unique solution on for any .*

*Proof. * From (18) and (19), we have, for , ,
Thus,
It follows from (29) that is a contraction. Thus, the conclusion of the theorem follows from the contraction mapping principle.

Theorem 7. * Let . Assume that meets the following condition: there exist , , such that
**
Then the problem (5) has unique solution on for
*

*Proof. *Let
where
By (18) and (19), we have, for ,
This, together with (36), yields that
Hence,
In view of , we have . Thus, by the following property of p-Laplacian operator:

if , , , then ; we have, for ,
Thus,
It follows from (33) that is a contraction. Thus, the conclusion of the theorem follows from the contraction mapping principle.

#### 4. Examples

*Example 8. * Consider the following boundary value problem:
where
Let
By computation, we deduce that
Thus, let ; we have
Hence, by Theorem 5, BVP (42) has at least one solution for .

*Example 9. *Consider the following boundary value problem:
where
Let
By computation, we deduce that
Let
Thus,