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: BOUNDEDNESS AND VANISHING OF SOLUTIONS FOR A FORCED DELAY DYNAMIC EQUATION | BOUNDEDNESS AND VANISHING OF SOLUTIONS FOR A FORCED DELAY DYNAMIC EQUATION DOUGLAS R. ANDERSON Received 30 March 2006 Revised 10 July 2006 Accepted 14 July 2006 We give conditions under which all solutions of a time-scale first-order nonlinear variable-delay dynamic equation with forcing term are bounded and vanish at infinity for arbitrary time scales that are unbounded above. A nontrivial example illustrating an application of the results is provided. Copyright 2006 Douglas R. Anderson. 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. Delay dynamic equation with forcing term Following Hilger s landmark paper 8 a rapidly expanding body of literature has sought to unify extend and generalize ideas from discrete calculus quantum calculus and continuous calculus to arbitrary time-scale calculus where a time scale is simply any nonempty closed set of real numbers. This paper illustrates this new understanding by extending some continuous results from differential equations to dynamic equations on time scales thus including as corollaries difference equations and q-difference equations. Throughout this work we consider the nonlinear forced delay dynamic equation xA t -p t f x t t r t t e to to t to 0 where T is a time scale unbounded above f R R is continuous and the functions p T 0 to and r T R are both right-dense continuous. Moreover the variable delay T T T is increasing with T t t for all t e t0 to t such that limt TO r t to. The initial function associated with takes the form x t y t for t e t ío t0 where y is rd-continuous on t t0 t0 . Equation is studied extensively by Qian and Sun 13 in the case when T R. See also related discussions on unforced delay equations by Matsunaga etal. 12 in the continuous case and by Erbe etal. 6 or Zhang and Yan 14 Hindawi Publishing Corporation Advances in
