The aim of this paper is to characterize the uniform exponential stability of difference equations. We obtain very general input–output conditions for stability of difference equations using diverse vector valued sequence spaces. As an application, we obtain an estimation for the lower bound of the stability radius of a linear control system of difference equations. Finally, we characterize the stability of systems of difference equations in terms of stabilizability and detectability, obtaining discrete-time versions for a result due to Clark, Latushkin, Montgomery-Smith and Randolph.

The aim of this paper is to obtain new criteria for the existence of the dichotomies of dynamical systems on the half-line. We associate to a discrete dynamical system an input-output system between two abstract sequence spaces. We deduce conditions for the existence of ordinary dichotomy and exponential dichotomy of the initial discrete system, by using certain admissibility properties of the associated input-output system. We establish the axiomatic structures of the input and output spaces, in each case, clarifying the underlying hypotheses as well as the generality of the proposed method. Next, we present a new and direct proof for the equivalence between the exponential dichotomy of an evolution family on the half-line and the exponential dichotomy of the associated discrete dynamical system. Finally, we apply our main results to the study of the exponential dichotomy of evolution families on the half-line.