Abstract
We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsive -difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.
1. Introduction
The subject of -calculus (also known as quantum calculus) is regarded as ordinary calculus without the idea of limit. The systematic development of -calculus started with the work of Jackson [1] at the beginning of the twentieth century. The application of -calculus covers a variety of topics such as special functions, particle physics and supersymmetry, combinatorics, initial and boundary value problems of -difference equations, operator theory, and control theory. For details of the advancement of -calculus, we refer the reader to the texts [2–4] and papers [5–13].
One of the advantages for considering -difference equations is that these equations are always completely controllable and appear in the -optimal control problem [14]. The variational -calculus is regarded as a generalization of the continuous variational calculus due to the presence of an extra parameter whose nature may be physical or economical. The variational calculus on the -uniform lattice includes the study of the -Euler equations and its applications to the isoperimetric and Lagrange problem and commutation equations. In other words, it suffices to solve the -Euler-Lagrange equation for finding the extremum of the functional involved instead of solving the Euler-Lagrange equation [15]. Further details can be found in [16–22].
Impulsive differential equations have extensively been studied in the past two decades. In particular, initial and boundary value problems of impulsive fractional differential equations have attracted the attention of many researchers; for instance, see [23–34] and references therein. In a recent work [32], the authors discussed the existence and uniqueness of solutions for impulsive -difference equations.
Motivated by [32], we investigate the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsive -difference equation in this paper. Precisely, we consider where are -derivatives , , , , , and , where and denote the right and the left limits of at , respectively.
Here, we remark that the classical -difference equations cannot be considered with impulses as the definition of -derivative fails to work when an impulse point for some . On the other hand, this situation does not arise for impulsive problems on -time scale as the points and are consecutive points. In quantum calculus on finite intervals, the points and are considered only in an interval . Therefore, the problems with impulses at fixed times can be considered in the framework of -calculus. For more details, see [32].
2. Preliminaries
Let us set and introduce the space as follows: with the norm . Then, is a Banach space.
Let us recall some basic concepts of -calculus [32].
For and , we define the -derivatives of a real valued continuous function as Higher order -derivatives are given by The -integral of a function is defined by provided that the series converges. If and is defined on the interval , then Observe that
Note that if and in (3) and (5), then and , where and are the well-known -derivative and -integral of the function defined by
Lemma 1. A function is a solution of the impulsive antiperiodic boundary value problem (1) if and only if it is a solution of the following impulsive -integral equation:
Proof. Let be a solution of (1). For , -integrating both sides of (1), we get Thus, we have For , -integrating both sides of (1), we obtain In view of , it follows that Similarly, we get Using the antiperiodic boundary value condition , we obtain (9). Conversely, assume that is a solution of the impulsive -integral equation (9); then by a direct computation, it follows that the solution given by (9) satisfies -difference equation (1). This completes the proof.
3. Main Results
Define an operator as Obviously, problem (1) is equivalent to a fixed point problem . In consequence, problem (1) has a solution if and only if the operator has a fixed point.
Theorem 2. Assume that there exist continuous functions and a nonnegative constant such that Then, problem (1) has at least one solution.
Proof. Let us denote and . Take and define . It is easy to verify that is a bounded, closed, and convex subset of .
In order to show that there exists a solution for problem (1), we have to establish that the operator has a fixed point in . The proof consists of two steps:
(i) .
For any , we have
which means . So, is . Consider the following:
(ii) the operator is relatively compact.
Let . For any with , we have
which is independent of and tends to zero as . Thus, is equicontinuous. Hence, is relatively compact as is uniformly bounded. Further, it is obvious that the operator is continuous in view of continuity of and . Therefore, the operator is completely continuous on . By the application of Schauder fixed point theorem, we conclude that the operator has at least one fixed point in . This, in turn, implies that problem (1) has at least one solution.
Theorem 3. Assume that there exist a function and a positive constant such that and for , and , and . Then, problem (1) has a unique solution.
Proof. For , we have
By the given condition, , it follows that . Therefore, is a contraction. By the contraction mapping principle, problem (1) has a unique solution.
4. Example
Example 1. Consider impulsive antiperiodic boundary value problem of nonlinear -difference equation: where , , , and . With , , and , it is easy to verify that all conditions of Theorem 2 hold. Thus, by Theorem 2, problem (21) has at least one solution.
Example 2. Consider impulsive antiperiodic boundary value problem of nonlinear -difference equation: where , , , and . With , , and , combining with and , it is easy to verify that all conditions of Theorem 3 hold. Thus, by Theorem 3, problem (22) has a unique solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (2012021002-3).