Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation: xn+1 = f (xn−r1 , . . . , xn−rk )g(xn−m1 , . . . , xn−ml )+ 1 f (xn−r1 , . . . , xn−rk )+ g(xn−m1 , . . . , xn−ml ) , n= 0,1, . . . , where f ∈ C((0,+∞)k, (0,+∞)) and g ∈ C((0,+∞)l, (0,+∞)) with k, l ∈ {1,2, . . .}, 0 r1 < · · ·< rk and 0 m1 < · · ·<ml , and the initial values are positive real numbers. We give sufficient conditions under which the unique equilibrium x = 1 of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the recent literature. © 2006 Elsevier Inc. All rights reserved.