On various multimap classes in the KKM theory and their applications

Sehei Park

doi:10.31197/atnaa.433869

Abstract

Fixed point theory of&nbsp;convex-valued multimaps are closely related to the KKM theory from the beginning. In the last twenty-five years, we introduced the acyclic multimap class, the admissible multimap class&nbsp; A_c^\kappa , the better admissible class&nbsp; B and the KKMadmissible classes KC, KO in the frame of the KKM theory.&nbsp;Our aim in this review is to collect the basic properties of ourmultimap classes and some mutual relations among them in general&nbsp;topological spaces or our abstract convex spaces. We add some newremarks and further comments to improve many of those results, and&nbsp;introduce some recent applications of our multimap classes.&nbsp;

Highlights

Since Kakutani obtained his celebrated fixed point theorem for convex-valued u.s.c. multimaps in 1941 and Eilenberg and Montgomerry extended it for acyclic maps in 1948, there have appeared many types of multimaps with applications in various fields in mathematics, economics, game theory, natural sciences, engineering, and others
Our aim in this review is to collect the basic properties of our multimap classes and some mutual relations among them in general topological spaces or our abstract convex spaces
In our previous work [43], we reviewed applications of our fixed point theorems for the multimap class of compact compositions of acyclic maps and, in [48], we collected most of fixed point theorems related to the KKM theory due to the author

Summary

Introduction

Since Kakutani obtained his celebrated fixed point theorem for convex-valued u.s.c. multimaps in 1941 and Eilenberg and Montgomerry extended it for acyclic maps in 1948, there have appeared many types of multimaps with applications in various fields in mathematics, economics, game theory, natural sciences, engineering, and others. In our previous work [43], we reviewed applications of our fixed point theorems for the multimap class of compact compositions of acyclic maps and, in [48], we collected most of fixed point theorems related to the KKM theory due to the author. The present review is an expanded version of our previous review given at a RIMS Workshop in 2017 [59]

CONVEX-VALUED MAPS

FAN-BROWDER MAPS

ACYCLIC MAPS

ADMISSIBLE MULTIMAP CLASS Aκc

BETTER ADMISSIBLE MULTIMAP CLASS B

BASIC THEOREMS IN THE KKM THEORY