Improving KKM Theory on General Metric Type Spaces

A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.

Fixed Points, Maximal Elements and Equilibria of Generalized Games in Abstract Convex Spaces

We firstly prove some new fixed point theorems for set-valued mappings in noncompact abstract convex space. Next, two existence theorems of maximal elements for class of𝒜C,θmapping and𝒜C,θ-majorized mapping are obtained. As in applications, we establish new equilibria existence theorems for qualitative games and generalized games. Our theorems improve and generalize the most known results in recent literature.

KKM Type Theorem and Ky Fan's Inequalities in Abstract Convex Spaces

In this paper, we first prove some type theorem based on the F-KKM mapping and generalized F-KKM mapping in abstract convex space. From this, we establishsome Ky Fan's inequalities.

Remarks on Weakly KKM Maps in Abstract Convex Spaces

A KKM space is an abstract convex space satisfying the KKM principle. We obtain variants of the KKM principle for KKM spaces related to weakly KKM maps and indicate some applications of them. These results properly generalize the corresponding ones inG-convex spaces andϕA-spaces(X,D;{ϕA}A∈〈D〉). Consequently, results by Balaj 2004, Liu 1991, and Tang et al. 2007 can be properly generalized and unified.