Exponential trichotomy for variational difference equations

Abstract

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of variational difference equations in terms of the solvability of an associated discrete-time system. First, we obtain the structure of the trichotomy projections. Then, we associate with a linear system of variational difference equations (A) an input–output system (S A ). The linear space of all sequences with finite support contained in Z + is denoted by ℱ(Z, X). We show that the uniform admissibility of the pair (ℓ∞(Z, X), ℱ(Z, X)) for the system (S A ) is a sufficient condition for the existence of uniform exponential trichotomy of the system (A). Next, we prove that the system (A) is uniformly exponentially trichotomic if and only if the pair (ℓ∞(Z, X), ℱ(Z, X)) is uniformly admissible for the associated input–output system (S A ). We also prove that the uniform exponential trichotomy of a linear skew-product flow is equivalent with the uniform exponential trichotomy of the variational difference equation associated with it. We apply our results to the study of the uniform exponential trichotomy of general linear skew-product flows in infinite-dimensional spaces.