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

Abstract

We consider a multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a rapidly oscillating (in time) right-hand side. Some summands on the right-hand side can be of a large amplitude proportional to the square root of the oscillation frequency. The Krylov–Bogolyubov averaging method is justified for this problem depending on a large parameter (high frequency oscillations). The passage to the limit is realized in this problem that is called perturbed in the Hölder space of vector functions on the time interval under consideration and we construct the (averaged) multipoint boundary value problem (i.e., we prove that the solutions to the perturbed and averaged problems are asymptotically close). The approach of this paper relies on the classical implicit function theorem in a Banach space and was firstly used by Simonenko for abstract parabolic equations in the case of the Cauchy problem and the problem of time-periodic solutions. The Krylov–Bogolyubov averaging method is most important, widely used, and developed rather fully for various classes of equations. The numerous articles on the systems of ordinary differential equations mainly study the Cauchy problem on a segment and the problems of periodic almost periodic and general solutions bounded on the entire time axis. However, boundary-value problems, especially multipoint boundary value problems, are still represented insufficiently in the literature.