The averaging method for weakly nonlinear operator equations

A method asymptotic with respect to a small parameter $\varepsilon$ is presented for solving Cauchy problems for the evolution equations
$$ u_t+Lu=\varepsilon f[u],\qquad u(0)=u_0, $$
where $L$ is a linear operator and $f$ is a nonlinear operator. It is assumed that the method of regular expansion in powers of $\varepsilon$ leads to secular terms. Such terms can be removed by suitably defining how the terms of the asymptotic solution depend on the slow variable $\tau=\varepsilon t$.
The proposed method is modified for equations of second order in $t$. The possibility of getting rid of the terms secular with respect to $\tau$, and of applying the asymptotic methods in the case of problems with strong forced resonance, is indicated. Examples are given which illustrate possibilities for the proposed methods.
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