Abstract and Applied Analysis

Volume 2014 (2014), Article ID 478185, 6 pages

http://dx.doi.org/10.1155/2014/478185

Research Article

Nonlocal Boundary Value Problem for Nonlinear Impulsive -Integrodifference Equation

Lihong Zhang,¹ Dumitru Baleanu,2,3,4 and Guotao Wang¹

¹School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

²Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

³Department of Mathematics, Faculty of Art and Sciences, 06530 Balgat, Turkey

⁴Institute of Space Sciences, R 76900 Magurele-Bucharest, Romania

Received 4 January 2014; Accepted 30 January 2014; Published 10 March 2014

Academic Editor: J. C. Prajapati

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.

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.

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.

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.

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.

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.

G. C. Wu and D. Baleanu, “Discrete fractional logistic map and its chaos,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 283–287, 2014.

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.

J. X. Sun, Nonlinear Functional Analysis and Its Application, Science Press, Beijing, China, 2008.