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

This paper is concerned with the following nonlinear difference equations x n + 1 = ∑ i = 1 l A s i x n − s i B + C ∏ j = 1 k x n − t j , n = 0 , 1 , … , where the initial data x − m , x − m + 1 , … , x − 1 , x 0 ∈ R + , m = max { s 1 , … , s l , t 1 , … , t k } , s 1 , … , s l , t 1 , … , t k are non-negative integers, and A s i , B , C are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium x ̄ = 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].