Research Article | Open Access

# Nonlocal Boundary Value Problem for Nonlinear Impulsive -Integrodifference Equation

**Academic Editor:**J. C. Prajapati

#### Abstract

A nonlinear impulsive integrodifference equation within the frame of -quantum calculus is investigated by applying using fixed point theorems. The conditions for existence and uniqueness of solutions are obtained.

#### 1. Introduction

Recently, by introducing and applying the fractional difference operators to real world problems (see, e.g., [1–7] and the references therein) we revitalized the importance of the quantum calculus [8]. However the real world phenomena are usually described by complex model based involving different types of operators. In this way we hope to understand deeper the dynamics of complex or hypercomplex systems and to reveal their hidden aspects.

On this line of thought in this paper, we study the existence and uniqueness of solutions for nonlinear -integrodifference equation with nonlocal boundary condition and impulses: where , are -derivatives and -integrals , respectively. , , , , , , , , where and denote the right and the left limits of at , respectively.

#### 2. Preliminaries

Let us set , , and introduce the space: with the norm . Then, is a Banach space.

For convenience, let us recall some basic concepts of -calculus [9].

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 the series converges. If and is defined on the interval , then Observe that For , the following reversing order of -integration holds

Note that if and in (3) and (5), then , , where and are the well-known -derivative and -integral of the function defined by

Lemma 1. *For given , the function is a solution of the impulsive -integrodifference equation
**
if and only if satisfies the -integral equation
*

*Proof. *Let be a solution of -difference equation (10). For , applying the operator on both sides of , we have
Thus,

Similarly, for , applying the operator on both sides of , then
In view of , it holds

Repeating the above process, we can get

Using the boundary value condition given in (10), it follows

Conversely, assume that satisfies the impulsive -integral equation (11); applying on both sides of (11) and substituting in (11), then (10) holds. This completes the proof.

#### 3. Main Results

Letting , in view of Lemma 1, we introduce an operator as By reversing the order of integration, we obtain Then, the impulsive -integrodifference equation (1) has a solution if and only if the operator equation has a fixed point.

In order to prove the existence of solutions for (1), we need the following known result [10].

Theorem 2. *Let be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in .*

Theorem 3. *Assume the following.**There exist nonnegative bounded functions such that
for any , .**There exist positive constants such that
for any , .**Then problem (1) has at least one solution provided
*

*Proof. *Firstly, we prove the operator is completely continuous. Clearly, continuity of the operator follows from the continuity of , , , and . Let be bounded. Then , ; there exist positive constants such that , , , . Thus
This implies .

Furthermore, for any , satisfying , we have
As , the right hand side of the above inequality tends to zero. Thus, is relatively compact. As a consequence of Arzela Ascoli's theorem, is a compact operator. Therefore, is a completely continuous operator.

Define the set .

Next, we show is bounded. Let ; then , . For any , by conditions and , we have
which implies
So, the set is bounded. Thus, Theorem 2 ensures the impulsive -integrodifference equation (1) has at least one solution.

Corollary 4. *Assume the following.**There exist nonnegative constants such that
* *for any , , .**Then problem (1) has at least one solution.*

Theorem 5. *Assume the following.**There exist nonnegative bounded functions and such that
* *for .**There exist positive constants such that
* *for and .**.**Then problem (1) has a unique solution.*

*Proof. *Denote , . For , by and , we have

As by , then . Therefore, is a contractive map. Thus, the conclusion of the Theorem 5 follows by Banach contraction mapping principle.

#### 4. Example

Consider the following nonlinear -integrodifference equation with impulses Obviously, , , + , , , and .

By a simple calculation, we can get

Take , , , , and . Then all conditions of Theorem 3 hold. By Theorem 3, nonlinear impulsive -integrodifference (31) has at least one 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

- K. S. Miller and B. Ross, “Fractional difference calculus,” in
*Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications*, Series in Mathematics & Its Applications, Horwood, Chichester, 1989, pp.139-152, Nihon University, Koriyama, Japan, May 1988. View at: Google Scholar - F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,”
*Proceedings of the American Mathematical Society*, vol. 137, no. 3, pp. 981–989, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. M. Atici and S. Senguel, “Modeling with fractional difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 369, no. 1, pp. 1–9, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Abdeljawad, D. Baleanu, F. Jarad, and R. P. Agarwal, “Fractional sums and differences with binomial coefficients,”
*Discrete Dynamics in Nature and Society*, vol. 2013, Article ID 104173, 6 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. A. Anastassiou, “Principles of delta fractional calculus on time scales and inequalities,”
*Mathematical and Computer Modelling*, vol. 52, no. 3-4, pp. 556–566, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. C. Wu and D. Baleanu, “Discrete fractional logistic map and its chaos,”
*Nonlinear Dynamics*, vol. 75, no. 1-2, pp. 283–287, 2014. View at: Publisher Site | Google Scholar - M. T. Holm,
*The Theory of Discrete Fractional Calculus: Development and Application [Ph.D. thesis]*, University of Nebraska-Lincoln, Lincoln, Nebraska, 2011. - V. Kac and P. Cheung,
*Quantum Calculus*, Springer, 2002. - J. Tariboon and S. K. Ntouyas, “Quantum calculus on finite intervals and applications to impulsive difference equations,”
*Advances in Difference Equations*, vol. 2013, no. 282, 2013. View at: Publisher Site | Google Scholar - J. X. Sun,
*Nonlinear Functional Analysis and Its Application*, Science Press, Beijing, China, 2008.

#### Copyright

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.