  We study conditions for the abstract linear functional differential equation
$\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions,
where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main
conditions are stated in terms of the spectrum of the monodromy operator
associated with the equation and the circular spectrum of the forcing term $f$.
The obtained results extend recent results on the subject.
