Admissible Banach function spaces and nonuniform stabilities

TL;DR

This paper introduces a new class of Banach function spaces tailored to nonuniform exponential bounds of evolution families, generalizes Datko's theorem, and provides novel criteria for nonuniform stability.

Contribution

It develops a new framework of Banach function spaces based on nonuniform behavior and extends classical stability theorems to this setting.

Findings

Generalized Datko's theorem for nonuniform spaces

Established new criteria for nonuniform stability

Connected nonuniform bounds with Banach function space norms

Abstract

For nonuniform exponentially bounded evolution families defined on Banach spaces, we introduce a class of Banach function spaces, whose norms are completely determined by the nonuniform behaviour of the corresponding evolution family. We generalize a classical theorem of Datko on these spaces. In addition, we obtain new criteria for the existence of nonuniform stability.