Averaging in Multifrequency Systems of Differential-Functional Equations

This work 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, 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.
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.