Abstract
For differential equations of the form in a Banach space a modification of the classical Krylov-Bogolyubov method is put forward. It allows complications in the construction of higher-order approximations which stem from the 'small denominators problem' to be avoided and many of the standard constraints on the behaviour of the function to be eliminated. The approach suggested is based on some results on the Fourier transforms of distributions.Bibliography: 17 titles.
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