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: ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NEUTRAL DYNAMIC EQUATIONS ON TIME SCALES | ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NEUTRAL DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Received 30 January 2006 Revised 17 March 2006 Accepted 17 March 2006 We investigate the boundedness and asymptotic behavior of a first-order neutral delay dynamic equation on arbitrary time scales extending some results from difference equations. 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. Neutral delay dynamic equation We consider on arbitrary time scales the neutral delay dynamic equation x t -p t x k t q t x e t 0 t e t0 o t where T is a time scale unbounded above the variable delays k Ể t0 oo t T are nondecreasing with k t Ể t t for all t e t0 oo t such that limt o k t Ể t o. The coefficient functions p q T R are right-dense continuous with p bounded and q 0. To clarify some notation take ỉ-1 t sup s ĩ s t ĩ- n 1 t ĩ-1 ĩ-n t for t e f t0 oo t and fn 1 t ĩ ĩn t for t e f-3 t0 oo t- For p and k above let o be the linear set of all functions given by o x T R x t -p t x k t A e Crd t0 o t R solutions of will belong to o. In the aftermath of Hilger s breakthrough paper 4 a rapidly diversifying body of literature has sought to unify extend and generalize ideas from discrete calculus continuous calculus and quantum calculus to arbitrary time-scale calculus where a time scale is merely a nonempty closed set of real numbers. This paper illustrates this new understanding by extending some discrete results from difference equations to dynamic equations on time scales. In particular is studied in 6 with T Z and p 0 and in 5 in the case when T Z with variable p. Much of the organization of and motivation for this paper arise from 5 6 . For more on delay dynamic equations see for Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 .
