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

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:

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][16][17][18][19]).
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 antiperiodic boundary value problem for higher-order fractional differential equations: where is the Caputo fractional derivative of order ; is a given continuous function.
If we take = = 1, = = 0 and = 2, then the problem (5) becomes the problem studied in [17]. In this paper, we let ̸ = 1. 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.

Background Materials and Preliminaries
Definition 1 (see [21]). The fractional integral of order with the lower limit 0 for a function is defined as where Γ is the gamma function.

Main Results
Let = 1 ([0, 1], ) denote the Banach space of continuous functions ( ) and ( ) from [0, 1] → endowed with the norm defined by where Define an operator F : → as From (18), we conclude that Then (5) Then the problem (5) has at least one solution on [0, ] for For ∈ , we have Discrete Dynamics in Nature and Society This, together with (21) and (22), yields that Hence, F( ) is uniformly bounded. Next we show that F is equicontinuous.
Thus, we conclude that F is equicontinuous on , and F : → is completely continuous.
By Schauder fixed point theorem we know that there exists a solution for the boundary value problem (5).
Proof. From (18) and (19), we have, for 1 , 2 ∈ , Discrete Dynamics in Nature and Society Discrete Dynamics in Nature and Society 7 Thus, It follows from (29) that F is a contraction. Thus, the conclusion of the theorem follows from the contraction mapping principle.
By computation, we deduce that Thus, It follows from Example 9 that