A note on the averaging principle for ordinary differential equations depending on the slow time

The work presents a “doubly” nonautonomous version of the averaging principle, applicable to equations that depend on a small parameter ɛ and on (fast) time τ , but also on slow time t = ɛ τ . The objectives are to establish optimal conditions on the dependence of the coefficients of the equations on t under which the averaging principle can be extended and to provide good estimates of the distance between the solutions of the initial equation and those of the averaged equation, always with τ varying in intervals of length proportional to 1 / ɛ . The applicability of these results is based on the fact that the estimates obtained are uniform with respect to the initial time at which the solutions of both equations coincide.