Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 603271 10 pages doi 2009 603271 Research Article Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale Eric R. Kaufmann Department of Mathematics Statistics University of Arkansas at Little Rock Little Rock AR 72204 USA Correspondence should be addressed to Eric R. Kaufmann erkaufmann@ Received 31 March 2009 Accepted 20 May 2009 Recommended by Victoria Otero-Espinar Let T be a periodic time scale with period p such that 0 ti T mp e T i 1 2 . n m e N and 0 ti ti 1. Assume each ti is dense. Using Schaeffer s theorem we show that the impulsive dynamic equation yA t -aft ya t f t y t t e T y tf y t- I ti y ti i 1 2 . n y 0 yfT where y t limt y t y fi y t- and yA is the A-derivative on T has a solution. Copyright 2009 Eric R. Kaufmann. 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. 1. Introduction Due to their importance in numerous application for example physics population dynamics industrial robotics optimal control and other areas many authors are studying dynamic equations with impulse effects see 1-19 and references therein. The primary motivation for this work are the papers by Kaufmann et al. 9 and Li et al. 12 . In 9 the authors used a fixed point theorem due to Krasnosel skil to establish the existence theorems for the impulsive dynamic equation yA t -a t yơ t f t y tf t e 0 T n T y tt yhi I ti y ti i 1 2 . n y 0 0 where y t limt t y t and yA is the A-derivative on T. 2 Advances in Difference Equations In 12 the authors gave sufficient conditions for the existence of solutions for the impulsive periodic boundary value problem equation u t Xut f tut u tk u t- Ik u tk k 1 2 . p u 0 u T where A e R A 0 T 0 and 0 to t1 tp tp 1 T. This paper extends and generalized the .