Nonlinear Multivalued Periodic Systems

Abstract

We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation F(t, x). We prove existence theorems for the “convex” problem (that is, F is convex valued and for the “nonconvex” problem (that is, F is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation F(t, x) is replaced by ext F(t, x), the extreme points of F(t, x)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.