Note on KKM maps and applications

Yq Chen,Jk Kim,Yj Cho,Bs Lee

doi:10.1155/fpta/2006/53286

Abstract

We apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. Some new results on Fan-Browder fixed point theorem, Fan's minimax theorem and coincidence theorem are obtained.

Highlights

In 1929, the KKM map was introduced by Knaster et al [13] and it provides the foundation for many well-known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan-Browder’s fixed point theorem, Nash’s equilibrium theorem, HartmanStampacchia’s variational inequality theorem and many others
The central idea of applying KKM theory to prove that a family of sets has nonempty intersection is to find a suitable space and a mapping defined on that space such that this mapping is a KKM mapping and the original family of sets has finite intersection property provided the resulted family of sets by this mapping has finite intersection property
We first introduce a large class of mappings that can be interpreted as KKM mappings, we apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem

Summary

Introduction

In 1929, the KKM map was introduced by Knaster et al [13] and it provides the foundation for many well-known existence results, such as Ky Fan’s minimax inequality theorem, Ky Fan-Browder’s fixed point theorem, Nash’s equilibrium theorem, HartmanStampacchia’s variational inequality theorem and many others (see [1, 2, 5,6,7,8,9,10,11,12, 14,15,16,17]). The central idea of applying KKM theory to prove that a family of sets has nonempty intersection is to find a suitable space and a mapping defined on that space such that this mapping is a KKM mapping and the original family of sets has finite intersection property provided the resulted family of sets by this mapping has finite intersection property Based this idea, we first introduce a large class of mappings that can be interpreted as KKM mappings, we apply the KKM technique to study fixed point theory, minimax inequality and coincidence theorem. If G : E → 2X is a closed valued map with the KKM property and there is a set G(x) such that G(x) is compact.

Fan-Browder’s fixed point theorem without compactness condition

Note on KKM maps and applications

Coincidence theorem and minimax theorem