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Báo cáo hóa học: "ON THE SYSTEM OF RATIONAL DIFFERENCE EQUATIONS xn+1 = f (xn , yn−k ), yn+1 = f (yn ,xn−k )"

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: ON THE SYSTEM OF RATIONAL DIFFERENCE EQUATIONS xn+1 = f (xn , yn−k ), yn+1 = f (yn ,xn−k ) | ON THE SYSTEM OF RATIONAL DIFFERENCE EQUATIONS xn 1 f xn yn-k yn 1 f yn xn-k TAIXIANG SUN HONGJIAN XI AND LIANG HONG Received 15 September 2005 Revised 27 October 2005 Accepted 13 November 2005 We study the global asymptotic behavior of the positive solutions of the system of rational difference equations x 1 f xn yn-k yn 1 f yn Xn-k n 0 1 2 . under appropriate assumptions where k e 1 2 . and the initial values x-k x-k 1 . x0 y-k y-k 1 . y0 e 0 to . We give sufficient conditions under which every positive solution of this equation converges to a positive equilibrium. The main theorem in 1 is included in our result. Copyright 2006 Taixiang Sun 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. Introduction Recently there has been published quite a lot of works concerning the behavior of positive solutions of systems of rational difference equations 2-7 . These results are not only valuable in their own right but they can provide insight into their differential counterparts. In 1 Camouzis and Papaschinopoulos studied the global asymptotic behavior of the positive solutions of the system of rational difference equations xn xn 1 1 y--- yn yn 1 1 x--- n 0 1 2 . 1.1 where k e 1 2 . and the initial values x-k x-k 1 . x0 y-k y-k 1 . y0 e 0 to . To be motivated by the above studies in this paper we consider the more general equation xn 1 f xn yn-k yn 1 f yn xn-k n 0 1 2 . 1.2 Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 16949 Pages 1-7 DOI 10.1155 ADE 2006 16949 2 System of rational difference equations where k e 1 2 the initial values x-k x-k 1 xo y-k y-k 1 yo e 0 o and f satisfies the following hypotheses. H1 f e C E X E 0 o with a inf u v eExEf u v e E where E e 0 o 0 o . H2 f u v is increasing in u and decreasing in v. H3 There exists a decreasing function g e C