This article confirmed by the Krylov-Bogolyubov averaging for systems of partial differential equations of the first order, containing oscillating in time with the frequency $\omega \gg 1$ terms, among which there are large proportional to $ \sqrt{\omega}$ with zero mean. Previously, for a narrower range of problems similar result was obtained in the joint work of the author of this article and V.B. Levenshtam. They considered the system of the same type, but do not depend on number equation coefficients. In this paper, it is a hard limit is removed. Upon confirmation of the averaging method used by the Krylov-Bogolyubov proved earlier by the author and V.B. Levenshtam averaging theorem Cauchy problem for systems of ordinary differential equations with large high-frequency terms, as well as the method of characteristics, allows to move from consideration of partial differential equations of the first order ordinary differential equations. Proven in this article theorem extends the theory of the averaging method Krylov-Bogolyubov to new classes of semilinear evolution systems of partial differential equations of the first order. Note that there are now quite a lot of work on the development of the theory of the averaging method for different classes of differential equations with large high-frequency terms.

Метод усреднения Крылова-Боголюбова обоснован для нового класса эволюционных систем полулинейных дифференциальных уравнений в частных производных первого порядка, содержащих осциллирующие с частотой $\omega\gg1$ слагаемые. При этом некоторые слагаемые пропорциональны $\sqrt{\omega}$.