We consider differential systems in ▫$\mathbb{R}^N$▫ driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation ▫$F(t, u, u')$▫. ...For periodic systems we prove the existence of extremal trajectories, that is solutions of the system in which ▫$F(t, u, u')$▫ is replaced by ▫$\text{ext}F(t, u, u')$▫ (= the extreme points of ▫$F(t, u, u')$)▫. For Dirichlet systems we show that the extremal trajectories approximate the solutions of the "convex" problem in the ▫$C^1(T, \mathbb{R}^N)$▫-norm (strong relaxation).
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