# A New Approach to Variational Inequalities of Parabolic Type

## TL;DR

This paper develops a new compactness theorem to enable the application of fixed point methods for solving fully nonlinear parabolic variational inequalities with time-dependent constraints.

## Contribution

It introduces a novel compactness theorem specifically designed for parabolic variational inequalities, facilitating the use of fixed point methods.

## Key findings

- Established a new compactness theorem for parabolic variational inequalities
- Applied fixed point method to prove weak solvability of nonlinear problems
- Extended analytical tools for time-dependent convex constraints

## Abstract

This paper is concerned with the weak solvability of fully nonlinear parabolic variational inequalities with time dependent convex constraints. As possible approaches to such problems, there are for instance the time-discretization method and the fixed point method of Schauder type with appropriate compactness theorems. In this paper, our attention is paid to the latter approach. However, there has not been prepared any appropriate compactness theorem up to date that enables us the direct application of fixed point method to variational inequalities of parabolic type. In order to establish it we have to start on the set up of a new compactness theorem for a wide class of parabolic variational inequalities.