Asymptotic periodic solutions of differential equations with infinite delay

TL;DR

This paper establishes conditions for the existence and uniqueness of asymptotic 1-periodic solutions in differential equations with infinite delay using spectral theory and evolution semigroups, with applications to a Lotka-Volterra model.

Contribution

It introduces new conditions for asymptotic periodic solutions in abstract differential equations with infinite delay, expanding understanding of long-term behaviors.

Findings

Proved existence and uniqueness of asymptotic 1-periodic solutions.

Applied results to a Lotka-Volterra model with diffusion and infinite delay.

Provided a framework for analyzing long-term periodic behaviors in delayed differential equations.

Abstract

In this paper, by using the spectral theory of functions and properties of evolution semigroups, we establish conditions on the existence, and uniqueness of asymptotic 1-periodic solutions to a class of abstract differential equations with infinite delay of the form \begin{equation*} \frac{d u(t)}{d t}=A u(t)+L(u_t)+f(t) \end{equation*} where $A$ is the generator of a strongly continuous semigroup of linear operators, $L$ is a bounded linear operator from a phase space $\mathscr{B}$ to a Banach space $X$, $u_t$ is an element of $\mathscr{B}$ which is defined as $u_t(\theta)=u(t+\theta)$ for $\theta \leq 0$ and $f$ is asymptotic 1-periodic in the sense that $\lim\limits_{t \rightarrow \infty}(f(t+1)-$ $f(t))=0$. A Lotka-Volterra model with diffusion and infinite delay is considered to illustrate our results.