A justification of the averaging method for abstract parabolic equations

In this paper the method of averaging of N. N. Bogoljubov is applied to abstract parabolic equations of the form
\begin{equation} \frac{dx}{dt}=Ax+f(x,\omega t), \end{equation}
where $A$ is a linear, in general unbounded, operator generating an analytic semigroup, and $f$ is an operator subordinate to $A$, in general a nonlinear map, possessing the mean
$$ \lim_{N\to+\infty}\frac1N\int_0^Nf(x,t)\,dt=Fx. $$
Other conditions on the mapping $f$ are formulated in terms of the theory of semigroups.
The main results are contained in two theorems.
Theorem 1 relates the initial value problem for equation (1) with the equation
\begin{equation} \frac{dy}{dt}=Ay+Fy. \end{equation}

Theorem 2, in the case of periodic dependence of the mapping $f$ on time, establishes a connection between the stability of the stationary solution to equation (2) and the stability of the corresponding periodic solution of (1).
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