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In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form $x\mapsto a{(x,x^{\text{'}})}^{\text{'}}$. In this problem the maximal monotone term is required to be defined everywhere in the state space ${ℝ}^N$. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form $x\mapsto {\left(a\left(x\right)x^{\text{'}}\right)}^{\text{'}}$. In this case the maximal...