Stability of a time discrete perturbed dynamical system with delay

Let C-n be the set of n complex vectors endowed with a norm parallel to.parallel to(Cn). Let A, B be two complex n x n matrices and tau a positive integer. In the present paper we consider the nonlinear difference equation with delay of the type u(k+1) = Au-k + Buk-tau + F-k(u(k), u(k-tau)), k = 0, 1, 2, . . . , where F-k : C-n x C-n --> C-n satisfies the condition parallel to F-k(x, y)parallel to(Cn) less than or equal to p parallel to x parallel to(Cn) + q parallel to y parallel to(Cn), k = 0, 1, 2, . . . , where p and q are positive constants. In this paper, absolute stability conditions for this equation are established.