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 B to a Banach space X, ut is an element of
B which is defined as ut(θ)=u(t+θ) for θ≤0
and f is asymptotic 1-periodic in the sense that t→∞lim(f(t+1)−f(t))=0. A Lotka-Volterra model with diffusion and infinite
delay is considered to illustrate our results.Comment: 13 page