A Bogolyubov-Type Theorem with a Nonconvex Constraint in Banach Spaces

Abstract

We prove an analogue of the classical Bogolyubov theorem, with a nonconvex constraint. In the case we consider, the constraint is the solution set of a Cauchy problem for a differential inclusion with a nonconvex right-hand side satisfying a Lipschitz condition. Our approach is based on a relaxation argument, as in the Filippov--Wazewski theorem.