  We establish an averaging principle on the real semi-axis for semi-linear
equation \begin{equation}\label{eqAb1} x'=\varepsilon (\mathcal A
x+f(t)+F(t,x))\nonumber \end{equation} with unbounded closed linear operator
$\mathcal A$ and asymptotically Poisson stable (in particular, asymptotically
stationary, asymptotically periodic, asymptotically quasi-periodic,
asymptotically almost periodic, asymptotically almost automorphic,
asymptotically recurrent) coefficients. Under some conditions we prove that
there exists at least one solution, which possesses the same asymptotically
recurrence property as the coefficients, in a small neighborhood of the
stationary solution to the averaged equation, and this solution converges to
the stationary solution of averaged equation uniformly on the real semi-axis
when the small parameter approaches to zero.
