Research Article | Open Access

# Impulsive Antiperiodic Boundary Value Problems for Nonlinear -Difference Equations

**Academic Editor:**Juan J. Nieto

#### 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).

#### References

- F. H. Jackson, “On $q$-difference equations,”
*American Journal of Mathematics*, vol. 32, pp. 305–314, 1910. View at: Publisher Site | Google Scholar - V. Kac and P. Cheung,
*Quantum Calculus*, Springer, 2002. - T. Ernst,
*A Comprehensive Treatment of q-Calculus*, Springer, 2012. - A. Aral, V. Gupta, and R. P. Agarwal,
*Applications of q-Calculus in Operator Theory*, Springer, 2013. - M. El-Shahed and H. A. Hassan, “Positive solutions of $q$-difference equation,”
*Proceedings of the American Mathematical Society*, vol. 138, no. 5, pp. 1733–1738, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Ahmad and S. K. Ntouyas, “Boundary value problems for $q$-difference inclusions,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 292860, 15 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Ahmad, S. K. Ntouyas, and I. K. Purnaras, “Existence results for nonlinear $q$-difference equations with nonlocal boundary conditions,”
*Communications on Applied Nonlinear Analysis*, vol. 19, pp. 59–72, 2012. View at: Google Scholar | Zentralblatt MATH - B. Ahmad, A. Alsaedi, and S. K. Ntouyas, “A study of second-order $q$-difference equations with boundary conditions,”
*Advances in Difference Equations*, vol. 2012, article 35, 2012. View at: Google Scholar - M. Bohner and R. Chieochan, “Floquet theory for $q$-difference equations,”
*Sarajevo Journal of Mathematics*, vol. 8, no. 21, pp. 355–366, 2012. View at: Publisher Site | Google Scholar - B. Ahmad and J. J. Nieto, “Basic theory of nonlinear third-order $q$-difference equations and inclusions,”
*Mathematical Modelling and Analysis*, vol. 18, pp. 122–135, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. Pongarm, S. Asawasamrit, and J. Tariboon, “Sequential derivatives of nonlinear $q$-difference equations with three-point $q$-integral boundary conditions,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 605169, 9 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. Area, E. Godoy, and J. J. Nieto, “Fixed point theory approach to boundary value problems for second-order difference equations on non-uniform lattices,”
*Advances in Difference Equations*, vol. 2014, article 14, 2014. View at: Google Scholar - I. Area, N. Atakishiyev, E. Godoy, and J. Rodal, “Linear partial q-difference equations on $q$-linear lattices and their bivariate $q$-orthogonal polynomial solutions,”
*Applied Mathematics and Computation*, vol. 223, pp. 520–536, 2013. View at: Google Scholar - G. Bangerezako, “$q$-Difference linear control systems,”
*Journal of Difference Equations and Applications*, vol. 17, no. 9, pp. 1229–1249, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Bangerezako, “Variational $q$-calculus,”
*Journal of Mathematical Analysis and Applications*, vol. 289, no. 2, pp. 650–665, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. D. Logan, “First integrals in the discrete variational calculus,”
*Aequationes Mathematicae*, vol. 9, no. 2-3, pp. 210–220, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Guermah, S. Djennoune, and M. Bettayeb, “Controllability and observability of linear discrete-time fractional-order systems,”
*International Journal of Applied Mathematics and Computer Science*, vol. 18, no. 2, pp. 213–222, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Bartosiewicz and E. Pawłuszewicz, “Realizations of linear control systems on time scales,”
*Control and Cybernetics*, vol. 35, no. 4, pp. 769–786, 2006. View at: Google Scholar | Zentralblatt MATH - D. Mozyrska and Z. Bartosiewicz, “On observability concepts for nonlinear discrete-time fractional order control systems,”
*New Trends in Nanotechnology and Fractional Calculus Applications*, vol. 4, pp. 305–312, 2010. View at: Google Scholar - T. Abdeljawad, F. Jarad, and D. Baleanu, “Vartiational optimal-control problems with delayed arguments on time scales,”
*Advances in Difference Equations*, vol. 2009, Article ID 840386, 2009. View at: Publisher Site | Google Scholar - F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, A. Carpinteri and F. Mainardi, Eds., Springer, New York, NY, USA, 1997. View at: Google Scholar - O. Agrawal, “Some generalized fractional calculus operators and their applications in integral equations,”
*Fractional Calculus and Applied Analysis*, vol. 15, pp. 700–711, 2012. View at: Publisher Site | Google Scholar - B. Ahmad and S. Sivasundaram, “Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,”
*Nonlinear Analysis: Hybrid Systems*, vol. 3, no. 3, pp. 251–258, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Tian and Z. Bai, “Existence results for the three-point impulsive boundary value problem involving fractional differential equations,”
*Computers and Mathematics with Applications*, vol. 59, no. 8, pp. 2601–2609, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 72, no. 3-4, pp. 1604–1615, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 74, no. 3, pp. 792–804, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Anti-periodic fractional boundary value problems,”
*Computers and Mathematics with Applications*, vol. 62, no. 3, pp. 1150–1156, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Fečkan, Y. Zhou, and J. R. Wang, “On the concept and existence of solution for impulsive fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, pp. 3050–3060, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. F. Zhou, S. Liu, and W. Jiang, “Complete controllability of impulsive fractional linear time-invariant systems with delay,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 374938, 7 pages, 2013. View at: Publisher Site | Google Scholar - G. Wang, B. Ahmad, L. Zhang, and J. J. Nieto, “Comments on the concept of existenc e of solution for impulsive fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, pp. 401–403, 2014. View at: Publisher Site | Google Scholar - A. Chauhan and J. Dabas, “Local and global existence of mild solution to an impulsive fractional functional integro-differential equation with nonlocal condition,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 19, pp. 821–829, 2014. View at: Publisher Site | Google Scholar - J. Tariboon and S. K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive difference equations,”
*Advances in Difference Equations*, vol. 2013, article 282, 2013. View at: Google Scholar - J. Tariboon and S. K. Ntouyas, “Boundary value problems for first-order impulsive functional $q$-integro- difference equations,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 374565, 2014. View at: Publisher Site | Google Scholar - L. Zhang, D. Baleanu, and G. Wang, “Nonlocal boundary value problem for nonlinear ${q}_{k}$ integro-difference equation,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 478185, 2014. View at: Publisher Site | Google Scholar

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Copyright © 2014 Lihong Zhang et al. 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.