# A Primal-Dual Approach of Weak Vector Equilibrium Problems

## TL;DR

This paper introduces a primal-dual framework for weak vector equilibrium problems in topological vector spaces, providing new existence conditions, duality results, and applications to perturbed problems, especially in reflexive Banach spaces.

## Contribution

It offers novel sufficient conditions for solutions, establishes duality results, and extends the analysis to perturbed problems without requiring compactness.

## Key findings

- New existence conditions for solutions in Hausdorff topological vector spaces.
- Duality results ensuring solution set coincidence for primal and dual problems.
- Applicability to perturbed vector equilibrium problems in reflexive Banach spaces.

## Abstract

In this paper we provide some new sufficient conditions that ensure the existence of the solution of a weak vector equilibrium problem in Hausdorff topological vector spaces ordered by a cone. Further, we introduce a dual problem and we provide conditions that assure the solution set of the original problem and its dual coincide. We show that many known problems from the literature can be treated in our primal-dual model. We provide several coercivity conditions in order to obtain solution existence of the primal-dual problems without compactness assumption. We pay a special attention to the case when the base space is a reflexive Banach space. We apply the results obtained to perturbed vector equilibrium problems.