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.
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