Dichotomy and bounded solutions of dynamical systems in the Hilbert space

TL;DR

This paper establishes necessary and sufficient conditions for the existence of bounded solutions in discrete dynamical systems within Hilbert spaces, especially addressing resonance cases and using concepts like exponential dichotomy and pseudoinvertibility.

Contribution

It introduces a comprehensive framework for analyzing bounded solutions in Hilbert space dynamical systems, including resonance cases, via exponential dichotomy and generalized invertibility.

Findings

Bounded solutions exist under specific exponential dichotomy conditions.

Resonance cases are characterized by generalized invertibility.

Application to systems with e-trichotomy demonstrates the framework's versatility.

Abstract

For a general discrete dynamics on a Banach and Hilbert spaces we give a necessary and sufficient conditions of the existence of bounded solutions under assumption that the homogeneous difference equation admits an exponential dichotomy on the semi-axes. We consider the so called resonance (critical) case when the uniqueness of solution is disturbed. We show that admissibility can be reformulated in the terms of generalised or pseudoinvertibility. As an application we consider the case when the corresponding dynamical system is e-trichotomy.