# A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results

## TL;DR

This paper introduces a comprehensive existence and uniqueness theory for a broad class of mixed variational-hemivariational inequalities in Banach spaces, employing advanced nonsmooth analysis and fixed point techniques.

## Contribution

It establishes the first general existence and uniqueness results for MVHVI problems without requiring operator compactness, using novel equivalence theorems and solution set properties.

## Key findings

- Proved a general existence theorem for MVHVI in Banach spaces.
- Demonstrated solution set properties: boundedness, convexity, closedness, and continuity.
- Established a uniqueness result under the LBB condition.

## Abstract

We investigate a generalized Lagrange multiplier system in a Banach space, called a mixed variational-hemivariational inequality (MVHVI, for short), which contains a hemivariational inequality and a variational inequality. First, we employ the Minty technique and a monotonicity argument to establish an equivalence theorem, which provides three different equivalent formulations of the inequality problem. Without compactness for one of operators in the problem, a general existence theorem for (MVHVI) is proved by using the Fan-Knaster-Kuratowski-Mazurkiewicz principle combined with methods of nonsmooth analysis. Furthermore, we demonstrate several crucial properties of the solution set to (MVHVI) which include boundedness, convexity, weak closedness, and continuity. Finally, a uniqueness result with respect to the first component of the solution for the inequality problem is proved by using the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. All results are obtained in a general functional framework in reflexive Banach spaces.