Uniform stability of nonlinear difference systems

We offer sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the nonlinear difference system x(n + 1) - lambda x(n) + f(n, x(n)) = 0, n is an element of N, where lambda is an element of [0, 1], f: N x S --> R, S is the set of all functions phi: N(-k, 0) --> R for some k is an element of N, and x(n) is an element of S is defined by x(n)(m) = x(n + m) for m is an element of N(-k, 0). (C) 1998 Academic Press.