Asymptotic Behavior of Solutions of periodic linear partial functional differential equations on the half line

TL;DR

This paper investigates the asymptotic behavior of solutions to periodic linear partial functional differential equations on the half line, focusing on conditions for asymptotic almost periodic solutions based on spectral properties.

Contribution

It extends existing results by establishing new spectral conditions for the existence of asymptotic almost periodic solutions in such equations.

Findings

Conditions for asymptotic almost periodic solutions are characterized by the spectrum of the monodromy operator.

Main results connect the circular spectrum of the forcing term with the solution behavior.

The work generalizes recent findings in the spectral analysis of functional differential equations.

Abstract

We study conditions for the abstract linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions, where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term $f$. The obtained results extend recent results on the subject.