Strong exponential dichotomy of discrete nonautonomous systems: Input-output criteria and strong dichotomy radius

Abstract

The aim of this paper is to provide a new study based on input-output methods for strong exponential dichotomy and to give complete characterizations for this notion in the uniform case. First, we point out the connections between strong exponential dichotomy and other dichotomy concepts for discrete nonautonomous systems on the whole line. After that, we prove that the projection families for strong exponential dichotomy are unique and we deduce their structure. Next, we consider a natural admissibility concept relative to an input-output system associated to the initial system and we show that, by working with input and output spaces in two general classes of sequence spaces, the admissibility properties considered herein ensure a full description of the strong dichotomic behavior, in the uniform case. Moreover, we obtain that when one deduces the strong exponential dichotomy via admissibility, the projections are uniformly bounded by a constant which depends on the fundamental functions of admissible spaces and on the input-output operator. We present a complete study, providing input-output criteria applicable to discrete nonautonomous systems, without any restriction on their coefficients and without assuming any type of growth for the associated propagator. As an application, we deduce an interval of variation for the strong dichotomy radius whose upper and lower bounds are expressed in terms of norms and spectral radii of certain input-output operators. Moreover, we present an illustrative example in which we determine the strong dichotomy radius by applying the central results in this paper.