We study perturbations of dynamical systems in Banach spaces for which time varies
on simple time scales consisting of families of isolated segments of the real axis. On a segment of the time scale, the system is governed by an ordinary differential equation; the transfer of a trajectory from a segment to the next one is determined by a map of the Banach space. The main problem which we study is the following one: given a trajectory of the original system, can we find a close trajectory of a perturbed system? We study perturbations applying the so-called multiscale approach: it is assumed that there exists a countable family of projections of the phase space and the smallness
conditions are imposed on the projections of perturbations. To find a solution close to a specified solution of the unperturbed system, we introduce a generalization of the Perron method.