Copyright © 2009 Meseret Tuba Gülpinar and Mustafa Bayram. 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

Our aim is to investigate the global behavior of the following fourth-order rational difference equation: xn+1=(xnxn−2xn−3+xn+xn−2+xn−3+a)/(xnxn−2+xnxn−3+xn−2xn−3+1+a), n=0,1,2,… where a∈[0,∞) and the initial values x−3,x−2,x−1,x0∈(0,∞). To verify that the positive equilibrium point of the equation is globally asymptotically stable, we used the rule of the successive lengths of positive and negative semicycles of nontrivial solutions of the aforementioned equation.