Variable Time-Step θ-Scheme for Nonlinear Second Order Evolution Inclusion

Abstract

We deal with a multivalued second order dynamical system involving a Clarke subdifferential of a locally Lipschitz functional. We apply a time discretization procedure to construct a sequence of solutions to a family of the approximate problems and show its convergence to a solution of the exact problem as the time step size vanishes. We consider a nonautonomous problem in which both the viscosity and the multivalued operators depend on time explicitly. The time discretization method we use, is the $\theta$-scheme with $\theta \in [\frac{1}{2}, 1]$, thus, in particular, the Crank-Nicolson scheme and the implicit Euler scheme are included. We apply our result to a class of dynamic hemivariational inequalities.