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 $x\mapsto a(x,x^{\prime })^{\prime }$. In this problem the maximal monotone term is required to be defined everywhere in the state space $\mathbb{R}^N$. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form $x\mapsto (a(x)x^{\prime })^{\prime }$. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.