The stability of the set of KKM points
type: Item
instance of: scholarly article
title: The stability of the set of KKM points
zbMATH Open document ID: 1029.54050
DOI: 10.1016/S0362-546X(03)00096-8
author: Shuwen Xiang
author: Jian Yu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-08-13T00:00:00Z
review text: Let \(X\) be a bounded complete convex subset of a normed linear space. A multivalued function \(G\) on \(X\) such that each \(G(x)\) is a compact nonempty subset of \(X\) is called a KKM mapping if, for every finite subset \(\{x_1,x_2,\dots, x_n\}\) of \(X\), its convex hull is contained in \(\bigcup^n_{i=1} G(x_i)\). Let \(M\) be a the set of KKM mappings on \(X\) and define a metric on \(M\) by setting \(\rho(G_1, G_2)= \sup_{x\in X} h(G_1(x), G_2(x))\) where \(h(G_1(x), G_2(x))\) denotes the Hausdorff distance between the compact sets \(G_1(x)\) and \(G_2(x)\). A theorem of Fan implies that the set \(F(G)= \bigcap_{x\in X} G(x)\), called the set of KKM points, is nonempty. A point \(x\in F(G)\) is essential if, given \(\varepsilon> 0\), there exists \(\delta> 0\) such that \(G'\in M\) and \(\rho(G, G')< \delta\), then there exists \(x'\in F(G')\) whose distance from \(x\) is less than \(\varepsilon\). A mapping \(G\in M\) is essential if all its KKM points are essential. The authors prove that there is a subset \(Q\) of \(M\) that is the countable intersection of open subsets of \(M\), and hence is dense in \(M\), that has the property that every mapping in \(Q\) is essential. Moreover, they show that for every \(G\in M\) there is component \(C(G)\) of \(F(G)\) that is essential. This means that, given \(\varepsilon> 0\), there exists \(\delta> 0\) such that if \(G'\in M\) with \(\rho(G, G')\), then \(F(G')\) intersects the \(\varepsilon\)-neighborhood of \(C(G)\) in \(X\).
reviewed by: Robert F. Brown
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54C60
zbMATH DE Number: 1963665
zbMATH Keywords: KKM point
zbMATH Keywords: Stability
zbMATH Keywords: Essential component
MaRDI profile type: Publication
cites work: Q5520521
cites work: Q3994935
cites work: Q5806961
cites work: Q5331764
cites work: Q5812718
cites work: Q3220809
cites work: On the Strategic Stability of Equilibria
cites work: The stability of coincident points for multivalued mappings
cites work: Essential equilibria of \(n\)-person noncooperative games
cites work: On essential components of the solution set of generalized games
cites work: On essential components of the set of Nash equilibrium points
cites work: Q4231657
full work available at URL: s0362-546x(03)00096-8
OpenAlex ID: W2003011299
description: scientific article; zbMATH DE number 1963665