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.