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Báo cáo sinh học: " Research Article Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation Bo Zheng"

Tuyển tập các báo cáo nghiên cứu về sinh học được đăng trên tạp chí sinh học Journal of Biology đề tài: Research Article Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation Bo Zheng | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 461014 17 pages doi 10.1155 2010 461014 Research Article Explicit Conditions for Stability of Nonlinear Scalar Delay Impulsive Difference Equation Bo Zheng College of Mathematics and Information Sciences Guangzhou University Guangzhou 510006 China Correspondence should be addressed to Bo Zheng zhengbo611@yahoo.com.cn Received 20 March 2010 Accepted 2 June 2010 Academic Editor Leonid Shaikhet Copyright 2010 Bo Zheng. 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. Sufficient conditions are obtained for the uniform stability and global attractivity of the zero solution of nonlinear scalar delay impulsive difference equation which extend and improve the known results in the literature. An example is also worked out to verify that the global attractivity condition is a sharp condition. 1. Introduction and Main Results Let R N and Z be the sets of real numbers natural numbers and integers respectively. For any a b e Z define Z a a a 1 . and Z a b a a 1 . . b when a b. It is well known that the theory of impulsive differential equations is emerging as an important area of investigation since it is not only richer than the corresponding theory of differential equations without impulse effects but also represents a more natural framework for mathematical modeling of many world phenomena 1 . Moreover such equations may exhibit several real-world phenomena such as rhythmical beating merging of solutions and noncontinuity of solutions. And hence ordinary differential equations and delay differential equations with impulses have been considered by many authors and numerous papers have been published on this class of equations and good results were obtained see e.g. 1-10 and references therein . Since the behavior of discrete .