The paper deals with the vector discrete dynamical system Xk+1 = AkXk + fk (xk). The well-known result by Perron states that this system is asymptotically stable if A k = A = const is stable and fk(x) f̃(x) = o(∥x∥). Perron's result gives no information about the size of the region of asymptotic stability and norms of solutions. In this paper, accurate estimates for the norms of solutions are derived. They give us stability conditions for (1.1) and bounds for the region of attraction of the stationary solution. Our approach is based on the "freezing" method for difference equations and on recent estimates for the powers of a constant matrix. We also discuss applications of our main result to partial reaction-diffusion difference equations.