Asymptotic periodic solutions of differential equations with infinite delay

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.