Justification of the Averaging Method for the Multipoint Boundary Value Problem on the Semi-axis for High-frequency ODE Systems with Large Terms

The Krylov–Bogolyubov averaging method is justified for a high-frequency normal system of ordinary differential equations containing terms proportional to the square root of the high frequency of oscillations with multipoint boundary conditions. Thus, for the initial (perturbed) non-autonomous problem depending on a large parameter, a corresponding autonomous problem independent of this parameter is constructed, and a limiting transition (averaging method) is justified. The latter means proving the asymptotic proximity of solutions to the perturbed and averaged problems, which is uniform in time.