Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 19618, 9 pages
doi:10.1155/2007/19618
Research Article
A Part-Metric-Related Inequality Chain and Application to the Stability Analysis of Difference Equation
College of Computer Science, Chongqing University, Chongqing 400044, China
Received 8 October 2006; Revised 13 December 2006; Accepted 14 December 2006
Academic Editor: Panayiotis D. Siafarikas
Copyright © 2007 Xiaofan Yang 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 find a new part-metric-related inequality of the form min{ai,1/ai:1≤i≤5}≤((1+w)a1a2a3+a4+a5)/(a1a2+a1a3+a2a3+wa4a5)≤max{ai,1/ai:1≤i≤5}, where 1≤w≤2. We then apply this result to show that c^=1
is a globally asymptotically stable equilibrium of the rational difference equation xn=(xn−1+xn−2+(1+w)xn−3xn−4xn−5)/(wxn−1xn−2+xn−3xn−4+xn−3xn−5+xn−4xn−5), n=1,2,…,a0,a−1,a−2,a−3,a−4>0.