tailieunhanh - Báo cáo hoa học: " Research Article On Boundedness of Solutions of the Difference Equation xn 1 pxn qxn−1 / 1 xn for q 1 p 1"

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 Boundedness of Solutions of the Difference Equation xn 1 pxn qxn−1 / 1 xn for q 1 p 1 | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 463169 11 pages doi 2009 463169 Research Article On Boundedness of Solutions of the Difference Equation xn 1 pxn qxn-1 1 xn for q 1 p 1 Hongjian Xi 1 2 Taixiang Sun 1 Weiyong Yu 1 and Jinfeng Zhao1 1 Department of Mathematics Guangxi University Nanning Guangxi 530004 China 2 Department of Mathematics Guangxi College of Finance and Economics Nanning Guangxi 530003 China Correspondence should be addressed to Taixiang Sun stx1963@ Received 4 February 2009 Revised 19 April 2009 Accepted 2 June 2009 Recommended by Agacik Zafer We study the boundedness of the difference equation xn 1 pxn qxn-1 1 xn n 0 1 . where q 1 p 1 and the initial values x-1 x0 e 0 x . We show that the solution xn -1 of this equation converges to x q p - 1 if xn x or xn x for all n -1 otherwise xn f -1 is unbounded. Besides we obtain the set of all initial values x-1 x0 e 0 x X 0 x such that the positive solutions xn f -1 of this equation are bounded which answers the open problem proposed by Kulenovic and Ladas 2002 . Copyright 2009 Hongjian Xi 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 this paper we study the following difference equation pxn qxn-1 xn 1 1 xn n 0 1 . where p q e 0 x with q 1 p and the initial values x-1 x0 e 0 x . The global behavior of for the case p q 1 is certainly folklore. It can be found for example in 1 see also a precise result in 2 . The global stability of for the case p q 1 follows from the main result in 3 see also Lemma 1 in Stevic s paper 4 . Some generalizations of Copson s result can be found for example in papers 5-8 . Some more sophisticated results such as finding the asymptotic behavior of solutions of for the case p q 1 even when p 0 can be found