Impulsive stabilization of delay difference equations and its application in Nicholson's blowflies model
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RESEARCH
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Impulsive stabilization of delay difference equations and its application in Nicholson’s blowflies model Kaining Wu* and Xiaohua Ding * Correspondence: kainingwu@163. com Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai, 264209, China
Abstract In this article, we consider the impulsive stabilization of delay difference equations. By employing the Lyapunov function and Razumikhin technique, we establish the criteria of exponential stability for impulsive delay difference equations. As an application, by using the results we obtained, we deal with the exponential stability of discrete impulsive delay Nicholson’s blowflies model. At last, an example is given to illustrate the efficiency of our results. Mathematics Subject Classification 2000: 39A30; 39A60; 39A10; 92B05. Keywords: impulsive, difference equation, exponential stability, stabilization, Nicholson’s blowflies model
Introduction Discrete systems exist in the word widely and most of them are described by the difference equations. The properties of difference equations, especially the stability and stabilization, were studied by many researchers, see [1-6] and the references therein. As well known, in the practice, many systems are subject to short-term disturbances, these disturbances are often described by impulses in the modeling process, therefore the impulsive systems arise in many scientific fields and there are many works were reported on impulsive systems [7-16]. In those works, the stability study for the impulsive system is one of the research focuses, see [11-16]. In the study of stability, the Lyapunov function and Razumikhin method were used by many authors, see, for example, [6,17]. In [6], the Razumikhin technique was extended to the discrete systems. Although the stability of impulsive delay difference equations has been studied in some articles, for example, see [18], there are few article concerning on impulsive stabilization of delay difference equations. From the article [19], we know that the continuity is crucial in the proof of the stabilization theorem under the continuous situation. However, under the discrete situation, there is no continuity to be utilized. The loss of continuity puts difficulties in the way to get the stabilization theorem. The main aim of this article is to establish the criteria of impulsive stabilization for delay difference equations, using the Lyapunov function and Razumikhin method. Biological models were studied by many authors, see [20-25] and the references therein. The stability of the positive equilibrium is a hot topic to be studied. In this © 2012 Wu and Ding; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wu and Ding Advances in Difference Equations 2012, 2012:88 http://www.advancesindifferenceequations.com/c
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