Advances in Difference Equations
Volume 2009 (2009), Article ID 463169, 11 pages
doi:10.1155/2009/463169
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

1Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Received 4 February 2009; Revised 19 April 2009; Accepted 2 June 2009

Academic Editor: Agacik Zafer

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.

Abstract

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∈(0,+∞). We show that the solution {xn}n=−1∞ of this equation converges to x¯=q+p−1 if xn≥x¯ or xn≤x¯ for all n≥−1; otherwise {xn}n=−1∞ is unbounded. Besides, we obtain the set of all initial values (x−1,x0)∈(0,+∞)×(0,+∞) such that the positive solutions {xn}n=−1∞ of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenović and Ladas (2002).