We consider a first order nonlinear periodic system with a maximal monotone term whose domain is not all of ▫$\mathbb{R}^N$▫ and a multivalued perturbation ▫$T(t, x)$▫. First we prove an existence ...theorem for the convex problem (that is, ▫$F$▫ is convex-valued). Then by strengthening the continuity on ▫$F(t, \cdot)$▫ we show the existence of extremal periodic solutions (that is, solutions passing from the extreme points of ▫$F(t, x)$)▫ and we prove a strong relaxation theorem (approximating the solutions of the convex problem, by certain extremal trajectories).
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