On Perturbed Weak Vector Equilibrium Problems under new Semi-continuities

TL;DR

This paper introduces a new weaker semicontinuity concept that guarantees solution existence for perturbed weak vector equilibrium problems, expanding applicability where traditional semicontinuities fail.

Contribution

It proposes a novel semicontinuity notion that is weaker than existing ones, ensuring closedness and solution existence in vector equilibrium problems.

Findings

New semicontinuity ensures closedness of key sets.

Provides sufficient conditions for solution existence.

Establishes dual problem relations.

Abstract

In this paper we introduce a new semicontinuity notion, which is weaker than upper semicontinuity, and assures the closedness of the sets $G(y)=\{x\in K: f(x,y)\not\in -\inte C\}.$ Furhter, this semicontinuity is also closed under addition. These two properties make our new semicontinuity applicable in situations where other semicontinuities, like quasi upper semicontinuity or order upper semicontinuity, fail. The above emphasized properties are some key tools in order to provide new sufficient conditions that ensure the existence of the solution of a perturbed 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 that every solution of the dual problem is also a solution of the perturbed weak vector equilibrium problem.