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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.
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