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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 308024, 12 pages
Generalized Antiperiodic Boundary Value Problems for the Fractional Differential Equation with p-Laplacian Operator
1Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, China
2Department 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.
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.
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 , 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 , 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 , 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 , 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 , , , .
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 ). The fractional integral of order with the lower limit for a function is defined as where is the gamma function.
Definition 2 (see ). Caputo's derivative of order with the lower limit for a function can be written as
Lemma 3 (see ). 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
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 .
Theorem 7. Let . Assume that meets the following condition: there exist , , such that Then the problem (5) has unique solution on for
By (18) and (19), we have, for ,
This, together with (36), yields that
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.
Example 9. Consider the following boundary value problem: where Let By computation, we deduce that Let Thus,