Solvability of strongly nonlinear boundary value problems for second order differential inclusions

In this paper we study a problem for a second order differential inclusion with Dirichlet, Neumann and mixed boundary conditions. The equation is driven by a nonlinear, not necessarily homogeneous, differential operator satisfying certain conditions and containing, as a particular case, the p$p$-Laplacian operator. We prove the existence of solutions both for the case in which the multivalued nonlinearity has convex values and for the case in which it has not convex values. The presence of a maximal monotone operator in the equation make the results applicable to gradient systems with non-smooth, time invariant, convex potential and differential variational inequalities.