Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion

An analogue of the classical theorem of Bogolyubov with non-convex constraint is proved. The constraint is the solution set of a differential inclusion with non-convex lower semicontinuous right-hand side. As an application we study the interrelation between the solutions of the problem of minimizing an integral functional with non-convex integrand on the solutions of the original inclusion and the solutions of the relaxation problem.