The rule of semicycle and global asymptotic stability for a fourth-order rational difference equation

摘要

In this paper, the rule for the lengths of positive and negative semicycles of nontrivial solutions of the following fourth-order rational difference equation,x(n+1) = x(n)(b)x(n-2) + x(n-3)(b) + a / x(n)(b) + x(n-2)x(n-3)(b) + a n = 0, 1, 2, ...,where a, b 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 successively occur is found to be..., 4(+), 3(-), 1(+), 2(-), 2(+), 1(-), 1(+), 1(-), 4(+), 3(-), 1(+), 2(-), 2(+), 1(-), 1(+), 1(-), 4(+), 3(-), 1(+), 2(-), 2(+), 1(-), 1(+), 1(-),..., by which the positive equilibrium point of the equation is verified to be globally asymptotically stable. (c) 2005 Elsevier Ltd. All rights reserved.