The Bogoljubov-Krylov averaging principle and the Cauchy problem for ordinary differential equations on the half-line

An existence and uniqueness theorem for the Cauchy problem for an ordinary differential equation on the half-line is proved under the hypothesis that the Cauchy problem for the averaged equation has a unique solution. A comparison between the exponential stability of the original equation and the averaged equation is also made. The results established below may be considered as anlogues of the classical Bogoljubov theorem for bounded solutions; they also provide a natural generalization of Mitropol'skij's averaging principle.