Second-order Differential Equations in Hilbert Space

We consider a nonlinear second-order differential inclusion with a convex-valued term and nonlinear multivalued boundary conditions. Using the Schauder fixed-point theorem and techniques from multivalued analysis and from monotone operators theories, we prove the existence of a solution. As a preliminary step, we established the existence, uniqueness and continuous dependence upon data of solutions to a second-order evolution equation. Examples of problems for optimization and PDE are also discussed.