# Convergence of a Time Discretization for Nonlinear Second Order Inclusion

## TL;DR

This paper proves the convergence of a time discretization scheme for a class of nonlinear second order inclusions, including applications to nonsmooth differential inclusions with Clarke subdifferentials.

## Contribution

It establishes the existence of solutions for nonlinear second order inclusions using a novel time discretization approach and applies it to nonsmooth, nonconvex problems involving Clarke subdifferentials.

## Key findings

- Weak convergence of approximate solutions to the exact solution
- Applicability to nonsmooth, nonconvex differential inclusions
- Demonstrated through two specific examples

## Abstract

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.