Averaging for high-frequency normal system of ordinary differential equations with multipoint boundary value problems

A multipoint boundary value problem for a nonlinear normal system of ordinary differential equations with a right-hand side rapidly oscillating in time is considered. Some terms on the right-hand side can have a large amplitude — proportional to the square root of the oscillation frequency. For this problem, which depends on a large parameter (high frequency of oscillations), the averaging method of Krylov–Bogolyubov is justified. Namely, for this problem, which is called perturbed, a limiting (averaged) multipoint boundary value problem is constructed and the passage to the limit (i. e., the asymptotic closeness of the solution of the perturbed and averaged problems is proved) in the Hölder space of vector functions defined on the considered time interval. The used approach in this paper is based on the classical implicit function theorem in Banach space; this approach in the theory of the averaging method was apparently first used by I. B. Simonenko (see the corresponding reference indicated in the article) when justifying this method for abstract parabolic equations in the case of the Cauchy problem and the problem of time-periodic solutions. The averaging method of Krylov–Bogolyubov is one of the most important asymptotic methods. It is widely known and developed with great completeness for various classes of equations. In numerous papers in which systems of ordinary differential equations are considered, mainly the Cauchy problem on an interval and the problems of periodic, almost periodic, and general solutions bounded on the entire time axis are studied. Boundary-value problems, especially multi-point boundary value problems, are still insufficiently represented in the literature.