Applications of the KKM property to coincidence theorems, equilibrium problems, minimax inequalities and variational relation problems

TL;DR

This paper extends coincidence theorems using the KKM property to non-convex correspondences, introduces properly quasi-convex-like correspondences, and applies these results to equilibrium problems, minimax inequalities, and variational relations.

Contribution

It introduces properly quasi-convex-like correspondences and applies the KKM principle to establish new existence theorems in non-convex settings.

Findings

Established coincidence-like theorems for non-convex correspondences.

Applied theorems to solve equilibrium problems and minimax inequalities.

Proved existence of solutions for generalized vector variational relation problems.

Abstract

In this paper, we establish coincidence-like results in the case when the values of the correspondences are not convex. In order to do this, we define a new type of correspondences, namely properly quasi-convex-like. Further, we apply the obtained theorems to solve equilibrium problems and to establish a minimax inequality. In the last part of the paper, we study the existence of solutions for generalized vector variational relation problems. Our analysis is based on the applications of the KKM principle. We establish existence theorems involving new hypothesis and we improve the results of some recent papers.