Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 432379, 17 pages
doi:10.1155/2010/432379
Research Article

Global Stability of a Rational Difference Equation

Guo-Mei Tang,1 Lin-Xia Hu,2 and Gang Ma1

1School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, China
2Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, China

Received 23 October 2010; Accepted 29 November 2010

Academic Editor: Manuel De la Sen

Copyright © 2010 Guo-Mei Tang 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 consider the higher-order nonlinear difference equation 𝑥 𝑛 + 1 = ( 𝑝 + 𝑞 𝑥 𝑛 − 𝑘 ) / ( 1 + 𝑥 𝑛 + 𝑟 𝑥 𝑛 − 𝑘 ) , 𝑛 = 0 , 1 , … with the parameters, and the initial conditions 𝑥 − 𝑘 , … , 𝑥 0 are nonnegative real numbers. We investigate the periodic character, invariant intervals, and the global asymptotic stability of all positive solutions of the above-mentioned equation. In particular, our results solve the open problem introduced by Kulenović and Ladas in their monograph (see Kulenović and Ladas, 2002).