Instability of discrete systems

In this paper, we give criteria for instability and asymptotic instability for the null solution to the non-autonomous system of difference equations $$ y(t+1)=A(t)y(t) + f(t,y(t)),quad f(t,0)=0,, $$ when the system $x(t+1)=A(t)x(t)$ is unstable. In particular for $A$ constant, we study instability from a new point of view. Our results are obtained using the method of discrete dichotomies, and cover a class of difference systems for which instability properties cannot be deduced from the classical results by Perron and Coppel.