The Bogoljubov-Krylov averaging principle and the Cauchy problem for ordinary differential equations on the half-line (Q1892162)

The authors give the justification of the averaging principle on the half-line \([0, + \infty)\) for the differential equation (1) \(dx/dt = f(t/ \varepsilon, x)\) considered in a Banach space \(X\). Here the function \(f(t,x)\) is bounded continuous and continuously Fréchet differentiable. Under some standard conditions on the linear part of \(f(t,x)\) the existence of a unique asymptotically stable solution for (1) is proved. An interesting approach to this problem based on the use of a nonclassical implicit function theorem is presented in the paper.