Global asymptotic stability of equilibrium point for a family of rational difference equations

This paper is concerned with the following nonlinear difference equations xn+1=∑i=1lAsixn−siB+C∏j=1kxn−tj,n=0,1,…, where the initial data x−m,x−m+1,…,x−1,x0∈R+, m=max{s1,…,sl,t1,…,tk}, s1,…,sl,t1,…,tk are non-negative integers, and Asi,B,C are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium x̄=0 of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references Cinar (2004) [6], Yang et al. (2005) [7] and Berenhaut et al. (2007) [8].