Existence of solutions for quasilinear second order differential inclusions with nonlinear boundary conditions

In this paper we consider a quasilinear second-order differential inclusion with a convex-valued multivalued term and nonlinear, multivalued boundary conditions. Using the Leray–Schauder fixed-point theorem and techniques from multivalued analysis and from nonlinear analysis, we prove the existence of a solution. Our formulation of the problem is general and includes as special cases the Dirichlet, the Neumann, the periodic problems, as well as certain Sturm–Liouville-type problems.