Bigun Y. Averaging in Multifrequency Systems of Differential-Functional Equations

The thesis is devoted to the development and justification of averaging circuit for systems of differential equations with slow and fast variables which pass through resonances in the course of evolution. In this work, the resonance relation for frequencies dependant on retardation of argument in fast variables is introduced. We build the uniform estimates for oscillation integrals that are appropriate to the multifrequency systems under the condition of invariable and variable delay. The case of systems with linear delay is considered in detail. Under imposed conditions the asymptotics of estimates is unimprovable. On the basis of obtained estimates, we prove new theorems on justification of averaging method for systems with invariable and variable delay when frequencies depend on time lag or slow variables. The case of systems with linearly transformed argument is studied. We propose and justify the averaging circuit for systems with delay when multipoint or integral boundary conditions are given. The averaging procedure is also applied to integral boundary conditions. In a small neighbourhood of solution of an averaged problem, we prove the existence and uniformity of solution or its existence if a vector of frequencies depends on slow variables. We justify the averaging circuits for systems of higher approximation if initial or boundary conditions are given. The problem of an infinite string oscillation is considered under the influence of multifrequency perturbations and the averaging method for it is given. The provided averaging circuits are demonstrated by model examples.