Abstract
A quasilinear system with the differentiation operator with respect to the directions of vector fields specified by Lyapunov’s system with respect to space independent variables and a multiperiodic system with respect to time variables is considered. We study the problem of the existence and uniqueness of a multiperiodic solution of a quasilinear system and we use methods of the theory of multiperiodic solutions of linear systems. The research partially reflects the multiperiodic structure of a solution of the initial problem for quasilinear systems. Conditions for the existence and uniqueness of a multiperiodic solution, an existence theorem of a solution of the initial problem, and the problem of multiperiodic solutions are given. They are proved by the method of contraction mappings defined on spaces of smooth functions.