Averaging for a High-Frequency Normal System of Ordinary Differential Equations with Multipoint Boundary Value Problems

We consider a multipoint boundary valueproblem for a nonlinear normal system of ordinary differential equationswith a rapidly oscillating (in time) right-handside. Some summands on the right-hand side can be ofa large amplitude proportional to the square root of the oscillation frequency.The Krylov-Bogolyubov averaging method is justifiedfor this problem depending on a large parameter (high frequency oscillations).The passage to the limit is realized in this problemthat is called perturbed in the Holder space of vector functionson the time interval under considerationand we construct the (averaged) multipoint boundary value problem(i.e., we prove that the solutions to theperturbed and averaged problems are asymptotically close).The approach of this paper relies on theclassical implicit function theorem in a Banach spaceand was firstly used by Simonenkofor abstract parabolic equations in the case ofthe Cauchy problem and the problem of time-periodic solutions.The Krylov-Bogolyubov averaging method is most important,widely used, and developed rather fullyfor various classes of equations.The numerous articles on the systems of ordinary differential equationsmainly study the Cauchy problem on a segment and the problems of periodicalmost periodic and general solutions bounded on the entire time axis.However, boundary-value problems,especially multipoint boundary value problems, are still representedinsufficiently in the literature.