Research Article | Open Access
Xianghu Liu, Yanfang Li, "Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations", Abstract and Applied Analysis, vol. 2014, Article ID 571536, 10 pages, 2014. https://doi.org/10.1155/2014/571536
Some Antiperiodic Boundary Value Problem for Nonlinear Fractional Impulsive Differential Equations
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.
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.  and Podlubny  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 ). 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.
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 .
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 ). 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.
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
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.
Example 1. Choose , , and , and consider the following fractional impulsive generalized antiperiodic boundary value problem: where
Example 2. Choose , , and ; consider the following fractional impulsive generalized antiperiodic boundary value problem: where