Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems

Abstract

The aim of this paper is to present a new and very general method for the study of the uniform exponential trichotomy of nonautonomous dynamical systems defined on the whole axis. We consider a discrete dynamical system and we introduce the property of \begin{document} $(r, p)$ \end{document} -admissibility relative to an associated control system, where \begin{document} $r, p∈ [1, ∞]$ \end{document} . In several constructive steps, we obtain full descriptions of the sufficient conditions and respectively of the necessary criteria for uniform exponential trichotomy based on the \begin{document} $(r, p)$ \end{document} -admissibility of the pair \begin{document} $(\ell^∞(\mathbb{Z}, X), \ell^1(\mathbb{Z}, X))$ \end{document} . In the same time, we provide a complete diagram of the \begin{document} $\ell^p$ \end{document} -spaces which can be considered in the admissible pairs for the study of the uniform exponential trichotomy of discrete dynamical systems. We present illustrative examples in order to motivate the hypotheses and the generality of our method. Finally, we apply the main results to obtain new criteria for uniform exponential trichotomy of dynamical systems modeled by evolution families using the admissibility of various pairs of \begin{document} $\ell^p$ \end{document} -spaces.