tailieunhanh - Báo cáo hóa học: "Research Article On a Conjecture for a Higher-Order Rational Difference Equation"

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 On a Conjecture for a Higher-Order Rational Difference Equation | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 394635 9 pages doi 2009 394635 Research Article On a Conjecture for a Higher-Order Rational Difference Equation Maoxin Liao 1 2 Xianhua Tang 1 and Changjin Xu1 3 1 School of Mathematical Sciences and Computing Technology Central South University Changsha Hunan 410083 China 2 School of Mathematics and Physics University of South China Hengyang Hunan 421001 China 3 College of Science Hunan Institute of Engineering Xiangtan Hunan 411104 China Correspondence should be addressed to Maoxin Liao maoxinliao@ Received 30 December 2008 Revised 11 March 2009 Accepted 14 March 2009 Recommended by Jianshe Yu This paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation xn n 1 x-n-k 1 n 1 x-n-k -1 n 1 x-n-k 1 - n 1 x-n-k -1 n 0 1 2 . where m is odd and x-km x-km 1 . x-1 e 0 to . Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al. 2007. Copyright 2009 Maoxin Liao 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 In 2007 Berenhaut et al. 1 proved that every solution of the following rational difference equation xn-k xn-m xn I .vv-------- n 0 1 2 . 1 xn-kxn-m converges to its unique equilibrium 1 where x-m x-m 1 . x-1 e 0 to and 1 k m. Based on this fact they put forward the following two conjectures. Conjecture . Suppose that 1 k l m and that xn satisfies xn-k xn-l xn-m xn-kxn-lxn-m xn n 1 xn-kxn-l xn-lxn-m xn-mxn-k 0 1 2 . with x-m x-m 1 . x-1 e 0 to . Then the sequence xn converges to the unique equilibrium 1. 2 Advances in Difference Equations Conjecture . Suppose that m is ođđ and 1 k1 k2 km and define S 1 2 . m .If xn satisfies xn f1 xn-k1 xn-k2 . . . xn-km f2