Asymptotic Behavior of Linear Almost Periodic Differential Equations

TL;DR

This paper investigates the long-term stability of solutions to non-autonomous linear differential equations with almost periodic coefficients in Banach spaces, providing a general stability criterion based on Perron conditions.

Contribution

It introduces a new stability condition for almost periodic linear differential equations using Perron conditions, extending existing stability analysis methods.

Findings

Established a general criterion for strong stability in almost periodic equations.

Connected stability conditions to Perron solvability criteria.

Enhanced understanding of asymptotic behavior in non-autonomous systems.

Abstract

The present paper is concerned with strong stability of solutions of non-autonomous equations of the form $\dot u(t)=A(t)u(t)$, where $A(t)$ is an unbounded operator in a Banach space depending almost periodically on $t$. A general condition on strong stability is given in terms of Perron conditions on the solvability of the associated inhomogeneous equation.