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 Asymptotic Constancy in Linear | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 789302 20 pages doi 2010 789302 Research Article Asymptotic Constancy in Linear Difference Equations Limit Formulae and Sharp Conditions Istvan Gyori and Laszlo Horvath Department of Mathematics University of Pannonia Egyetem u. 10. Veszprem 8200 Hungary Correspondence should be addressed to Istvan Gyori gyori@ Received 20 January 2010 Accepted 23 March 2010 Academic Editor Agacik Zafer Copyright 2010 I. Gyori and L. Horvath. 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. It is found that every solution of a system of linear delay difference equations has finite limit at infinity if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices our conditions are also necessary. The novelty of our results is illustrated by examples. 1. Introduction Consider the nonautonomous linear delay difference system m y n 1 - y n 2Ai n y n - Ti n - y n - ơi n n 0 i 1 where the following are considered. A1 m 1 is an integer and A-ifn e Rdxd 1 i m n 0 . A2 Ti n n 0 and ơi n n 0 are sequences of nonnegative integers 1 i m such that s max max supT-ifn supơi ù 1 i m n 0 n 0 is finite. 2 Advances in Difference Equations Without loss of generality we may and do assume the following. A3 For each 1 i m and n 0 Ti n ơi ri 0 T1 n Tm n . Under these conditions s max1 i m maxn 0ơi n . Whenever the delays are constants we get the system y n 1 - y n Ai n y n - ki - y n - li Ỵ n 0 i 1 where we suppose that A4 ki li 1 i m are nonnegative integers and 0 k1 km. In this case s max1 i m l1 . lm . Together with the above equations we assume initial conditions of the
