Stability of Solutions for a Family of Nonlinear Difference Equations

We consider the family of nonlinear difference equations: xn 1 ∑ 3 i 1fi xn, . . . , xn−k f4 xn, . . . , xn−k f5 xn, . . . , xn−k / f1 xn, . . . , xn−k f2 xn, . . . , xn−k ∑ 5 i 3fi xn, . . . , xn−k , n 0, 1, . . . , where fi ∈ C 0, ∞ k , 0, ∞ , for i ∈ {1, 2, 4, 5}, f3 ∈ C 0, ∞ k , 0, ∞ , k ∈ {1, 2, . . . } and the initial values x−k, x−k 1, . . . , x0 ∈ 0, ∞ . We give sufficient conditions under which the unique equilibrium x 1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.