Coincidence theorems and minimax inequalities in abstract convex spaces

Yeol Je Cho,Mohsen Rostamian Delavar,Seyed Asghar Mohammadzadeh,Mehdi Roohi

doi:10.1186/1029-242x-2011-126

Abstract

Abstract In this paper, we deal with the notion of abstract convex spaces via minimal spaces as an extended version of other forms of convexity and establish some well-known results such as coincidence theorems for the classes m-KKM and ms-KKM of multimaps and Ky Fan's type minimax inequality. Mathematics Subject Classification (2000) 26A51; 26B25; 54H25; 55M20; 47H10; 54A05.

Highlights

Many problems in nonlinear analysis can be solved by showing the nonemptyness of the intersection of certain family of subsets of an underlying set
The first remarkable result on the nonempty intersection was the celebrated KnasterKuratowski-Mazurkiewicz theorem (KKM principle) in 1929 [1], which concerns with certain types of multimaps called the KKM maps later
The KKM theory first called by Park in [2] and [3] was the study of KKM maps and their applications

Summary

Introduction

Many problems in nonlinear analysis can be solved by showing the nonemptyness of the intersection of certain family of subsets of an underlying set. Zafarani [18] and Park [19] have introduced a new concept of abstract convex space and certain broad classes KC and KO of multimaps (having the KKM property) With this new concept, the KKM type maps were used to obtain matching theorems, coincidence theorems, fixed point theorems and others. Motivated by the results of Chang et al [23], we introduce a new definition about the family of multimaps with the ms-KKMC (ms-KKMO) property in minimal abstract convex space as follows. Let Y be a nonempty set, Z be a minimal space, s : Y ® D be a function and (X, D, Γ) be an abstract convex space. B ⊆ Gi ∪ m - Int Ac ⊆ Gi ∪ Uj : Uj ∈ M, Uj ⊆ Ac

Thus the compactness of B implies that n m n

There exists y’