Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms

Abstract

We consider differential systems in RN 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 extF(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 C1(T,RN)-norm (strong relaxation).