Advances in Difference Equations
Volume 2009 (2009), Article ID 394635, 9 pages
doi:10.1155/2009/394635
Research Article

On a Conjecture for a Higher-Order Rational Difference Equation

Maoxin Liao,1,2 Xianhua Tang,1 and Changjin Xu1,3

1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China
2School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China
3College of Science, Hunan Institute of Engineering, Xiangtan, Hunan 411104, China

Received 30 December 2008; Revised 11 March 2009; Accepted 14 March 2009

Academic Editor: Jianshe Yu

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.

Abstract

This paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation xn=(∏j=1m(xn−kj+1)+∏j=1m(xn−kj−1))/(∏j=1m(xn−kj+1)−∏j=1m(xn−kj−1)), n=0,1,2,…, where m is odd and x−km,x−km+1,…,x−1∈(0,∞). Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.