Global Asymptotic Stability for a Fourth-Order Rational Difference Equation

Abstract

Our aim is to investigate the global behavior of the following fourth-order rational difference equation: x(n+1) = (x(n)x(n-2)x(n-3) + x(n) + x(n-2) + x(n-3) + a) / (x(n)x(n-2) + x(n)x(n-3) + x(n-2)x(n-3) + 1 + a), n = 0, 1, 2, ... where a is an element of [0,infinity) and the initial values x(-3), x(-2), x(-1), x(0) is an element of (0, infinity). 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. Copyright (C) 2009 M. T. Gulpinar and M. Bayram.