Global Asymptotic Stability of a Family of Nonlinear Difference Equations

Abstract

In this note, we consider global asymptotic stability of the following nonlinear difference equationxn=(∏i=1v(xn-kiβi+1)+∏i=1v(xn-kiβi-1))/(∏i=1v(xn-kiβi+1)-∏i=1v(xn-kiβi-1)),  n=0,1,…, whereki∈ℕ  (i=1,2,…,v),  v≥2,β1∈[-1,1],β2,β3,…,βv∈(-∞,+∞),x-m,x-m+1,…,x-1∈(0,∞), andm=max1≤i≤v{ki}. Our result generalizes the corresponding results in the recent literature and simultaneously conforms to a conjecture in the work by Berenhaut et al. (2007).