Integral Equations and Exponential Trichotomy of Skew-Product Flows

We are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skew-product flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skew-product flows in infinite-dimensional spaces, using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations, the qualitative theory of dynamical systems in continuous time, and certain related control problems.