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: DELAY DYNAMIC EQUATIONS WITH STABILITY | DELAY DYNAMIC EQUATIONS WITH STABILITY DOUGLAS R. ANDERSON ROBERT J. KRUEGER AND ALLAN C. PETERSON Received 13 August 2005 Accepted 23 October 2005 We first give conditions which guarantee that every solution of a first order linear delay dynamic equation for isolated time scales vanishes at infinity. Several interesting examples are given. In the last half of the paper we give conditions under which the trivial solution of a nonlinear delay dynamic equation is asymptotically stable for arbitrary time scales. Copyright 2006 Douglas R. Anderson 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. 1. Preliminaries The unification and extension of continuous calculus discrete calculus -calculus and indeed arbitrary real-number calculus to time-scale calculus where a time scale is simply any nonempty closed set of real numbers were first accomplished by Hilger in 4 . Since then time-scale calculus has made steady inroads in explaining the interconnections that exist among the various calculuses and in extending our understanding to a new more general and overarching theory. The purpose of this work is to illustrate this new understanding by extending some continuous and discrete delay equations to certain time scales. Examples will include specific cases in differential equations difference equations -difference equations and harmonic-number equations. The definitions that follow here will serve as a short primer on the time-scale calculus they can be found in 1 2 and the references therein. Definition . Define the forward backward jump operator Ơ t at t for t sup T resp. p t at t for t inf T by Ơ t inf t t T e T p t sup T t T e T Vt e T. Also define Ơ sup T sup T if sup T TO and p inf T inf T if inf T -TO. Define the graininess function p T R by p t Ơ t - t. Hindawi Publishing Corporation Advances .
