Geodesic
The averaging method for weakly nonlinear operator equations
Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 223-242 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. Bibliography: 16 titles. 
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     title = {The averaging method for weakly nonlinear operator equations},
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A. L. Štaras. The averaging method for weakly nonlinear operator equations. Sbornik. Mathematics, Tome 62 (1989) no. 1, pp. 223-242. http://geodesic.mathdoc.fr/item/SM_1989_62_1_a14/

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