Discrete Dynamics in Nature and Society
Volume 2009 (2009), Article ID 616982, 7 pages
Research Article

Global Asymptotic Stability for a Fourth-Order Rational Difference Equation

1Department of Mathematics, Faculty of Arts and Sciences, Yıldız Technical University, 34210 Esenler, İstanbul, Turkey
2Department of Mathematics, Faculty of Arts and Sciences, Fatih University, 34500 Buyukcekmece, İstanbul, Turkey

Received 8 May 2009; Revised 4 June 2009; Accepted 4 June 2009

Academic Editor: Elena Braverman

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.


Our aim is to investigate the global behavior of the following fourth-order rational difference equation: xn+1=(xnxn2xn3+xn+xn2+xn3+a)/(xnxn2+xnxn3+xn2xn3+1+a), n=0,1,2, where a[0,) and the initial values x3,x2,x1,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.