# Time-dependent Inclusions and Sweeping Processes in Contact Mechanics

## TL;DR

This paper establishes existence and uniqueness results for a class of time-dependent inclusions and sweeping processes in contact mechanics, with applications to viscoelastic contact problems involving deformable bodies.

## Contribution

It introduces a new theoretical framework for analyzing time-dependent inclusions and sweeping processes with velocity constraints, proving their unique weak solvability.

## Key findings

- Proved existence and uniqueness of solutions for the new class of inclusions.
- Applied abstract results to specific viscoelastic contact problems.
- Demonstrated the relevance of the theory to models of deformable bodies in contact.

## Abstract

We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then we use this result to prove the unique weak solvability of a new class of Moreau's sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.