Almost periodic solutions of periodic linear partial functional differential equations

TL;DR

This paper investigates conditions under which periodic linear functional differential equations have almost periodic solutions sharing the same frequency structure as the forcing term, extending recent theoretical results.

Contribution

It provides new spectral conditions involving the monodromy operator for the existence of almost periodic solutions in periodic linear functional differential equations.

Findings

Conditions based on the spectrum of the monodromy operator ensure almost periodic solutions.

Results extend previous work by relaxing certain assumptions.

Discussion on potential extensions to time-dependent operators A(t).

Abstract

We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.