# Existence Result for Generalized Variational Equality

## TL;DR

This paper establishes the existence of solutions for generalized variational inequalities under weaker conditions, extending previous results and addressing an open problem in the field.

## Contribution

It provides new existence theorems for generalized variational inequalities with relaxed assumptions and extends prior results using generalized continuity and monotonicity.

## Key findings

- Proved existence of solutions under weakened assumptions
- Extended previous existence results for generalized equations
- Addressed an open problem in variational inequality theory

## Abstract

In this paper, we prove the existence of a solution to the Stampachia variational inequality under weakened assumptions on the given operator. As a consequence, we provide some sufficient conditions that under them the generalized equation $0\in T(x)$ has a solution. Furthermore, by using generalized results of continuity and monotonicity, we extend the related existence results and we answer an open problem proposed by Kassay and Miholka (J Optim Theory Appl 159 (2013) 721-740).