Skutar I. Asymptotic integration of systems of differential equations with a small parameter at some derivatives.

The thesis is dedicated to the development and substantiation of the averaging method for systems of differential equations with slow and fast variables and linearly transformed arguments, for which frequency resonance conditions are fulfilled in the process of evolution, as well as to the construction of asymptotic and of global solutions of systems of linear equations with a small parameter at some derivatives. Sufficient solvability conditions for multifrequency systems with local-integral conditions are established. The method of averaging over fast variables is substantiated and the accuracy estimations of the averaging method are constructed, which obviously depends on the small parameter. The results are illustrated by model examples. An asymptotic solution of a system of linear differential equations with a small parameter for some derivatives and coefficients depending on this parameter is constructed by reducing this system to a simpler form. The global solutions of systems of differential equations with a deviating argument and a small parameter are investigated, and the formal solution of the integrodifferential equation, which is obtained by asymptotic integration of one system of linear differential equations with a small parameter at some derivatives, is constructed. The initial problem for the hyperbolic equation on the axis under the action of multifrequency perturbations by a system with linearly transformed arguments is considered. Its solvability is proved, the method of averaging is substantiated and illustrated by a model example.