The paper deals with the vector discrete dynamical system
$x_{k+1}=A_{k}x_{k}+f_{k}( x_{k})$ . The well-known result by
Perron states that this system is asymptotically stable if
$A_{k}\equiv A=\const$
is stable and $f_{k}(x) \equiv \tilde f(x) =o( \| x\| )$ . 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.