Strongly nonlinear multivalued systems involving singular $Phi$-Laplacian operators

In this paper we study two vector problems with homogeneous Dirichlet boundary conditions for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is governed by a nonlinear differential operator of the form x bar right arrow(k(t)Phi(x'))', where k is an element of C(T, IR(+)) and Phi is an increasing homeomorphism defined on a bounded domain. In this problem the maximal monotone term need not be defined everywhere in the state space R(N), incorporating into our framework differential variational inequalities. The second problem is governed by the more general differential operator of the type x bar right arrow (a(t,x)Phi(x'))', where a(t,x) is a positive and continuous scalar function. In this case the maximal monotone term is required to be defined everywhere.