Admissibility approach to nonuniform exponential dichotomies roughness with nonlocal perturbations

TL;DR

This paper investigates how nonuniform exponential dichotomies are preserved under nonlocal perturbations using admissibility of function classes, providing conditions that ensure stability of these dichotomies.

Contribution

It introduces a new approach using admissibility to analyze the robustness of nonuniform exponential dichotomies under nonlocal perturbations.

Findings

Established sufficient conditions for preservation of dichotomies

Identified smallness integrability condition for stability

Extended the understanding of nonuniform hyperbolicity robustness

Abstract

Nonuniform exponential dichotomy serves as an important characteristic of nonuniform hyperbolicity, while admissibility of function classes is often used to characterize nonuniform exponential dichotomy. In this paper, we investigate the preservation of nonuniform exponential dichotomy under certain nonlocal perturbations. By utilizing the concept of admissibility of a pair of function classes, we establish sufficient conditions to ensure that the dichotomy results are consistent with those in the homogeneous situation. These results need to satisfy a smallness integrability condition.