KKM lemmas and minimax inequality theorems in abstract convexity spaces

Shuwen Xiang,Shunyou Xia,Jing Chen

doi:10.1186/1687-1812-2013-209

Shuwen Xiang, Shunyou Xia + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2013-209

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Abstract

The purpose of this paper is to give some further results in a type of generalized convexity spaces. First, we prove that an abstract convex space has KKM property if and only if it has a strong Fan-Browder property. Then we introduce an abstract convex structure via an upper semi-continuous multi-valued mapping and establish some generalized versions of KKM lemma. By employing our general KKM lemmas, we derive some generalizations of minimax inequalities, which contain several existing ones as special cases.

Highlights

1 Introduction Many problems in nonlinear analysis can be solved by nonempty intersection of a certain family of subsets of an underlying set
We prove that an abstract convex space has a KKM property if and only if it has a strong Fan-Browder property
Let (Y, C) be an abstract convexity space, and let X be a subset of Y

Summary

Introduction

Many problems in nonlinear analysis can be solved by nonempty intersection of a certain family of subsets of an underlying set. We prove that an abstract convex space has a KKM property if and only if it has a strong Fan-Browder property. Let (Y , C) be an abstract convexity space, let X be a subset of Y , and let F : X → X be a multi-valued mapping. (iii) F : X → X is said to be a weak Fan-Browder mapping if F is weakly convex- valued and has relatively open preimages in X. (iii) If F : X → X is a weak Fan-Browder mapping with no fixed point, T : X → X is a KKM mapping with closed values.

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