Convergence of a Time Discretization for Nonlinear Second Order Inclusion

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multivalued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second order in time differential inclusion involving Clarke subdifferential of a locally Lipschitz, possibly nonconvex and nonsmooth potential. In two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.