Generalized KKM-type theorems in GFC-spaces and applications
type: Item
instance of: scholarly article
title: Generalized KKM-type theorems in GFC-spaces and applications
zbMATH Open document ID: 1176.47041
author: Phan Quoc Khanh
author: Nguyen Hong Quan
author: Jen-Chih Yao
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-06-24T00:00:00Z
review text: A triplet \((X, Y, \Phi )\) is called a generalized finitely continuous topological space (\(GFC\)-space in short) if \(X\) is a topological space, \(Y\) is a nonempty set and \(\Phi\) is a family of continuous function \(\phi:\Delta_n \rightarrow X\), \(n\in \mathbb{N}\), such that to each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\) is associated a function \(\phi_N:\Delta_n \rightarrow X\) from the family \(\Phi\). Let \((X, Y, \Phi )\) be a \(GFC\)-space, \(Z\) be a topological space, \(F: Y\rightarrow 2^Z\) and \(T:X \rightarrow 2^Z\). \(F\) is said to be a \(T-KKM\) mapping if, for each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\) and each \(\{y_{i_0},\dots, y_{i_k}\}\subseteq N\), one has \(T(\phi_N(\Delta_k))\subseteq \bigcup_{j=0}^k F(y_{i_j})\). A set-valued mapping \(T:X \rightarrow 2^Z\) has the generalized \(KKM\) property if, for each \(T-KKM\) mapping \(F: Y\rightarrow 2^Z\), the family \(\{ \overline{F(y)}: y\in Y\}\) has the finite intersection property. The main result of the present paper is the following \(KKM\)-type theorem: If \(T: X \rightarrow 2^Z\), \(F:Y\rightarrow 2^Z\) and \(S:Y\rightarrow 2^X\), then \(\overline{T(S(Y))}\cap \bigcap_{\in Y} F(y)\neq \emptyset\) whenever the following conditions are satisfied: (i) for each finite subset \(N=\{ y_0,\dots, y_n\}\) of \(Y\), \(\phi_N(\Delta_k)\subseteq S(Y)\). (ii) \(\overline{T(S(Y))}\) is compact. (iii) \(T\) has the generalized \(KKM\) property. (iv) \(F\) is \(T-KKM\) and transfer compactly closed-valued.
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 55M20
zbMATH DE Number: 5569417
zbMATH Keywords: GFC-spaces
zbMATH Keywords: \(T\)-KKM mappings
zbMATH Keywords: generalised KKM property
zbMATH Keywords: transfer compactly closed-valued (open-valued) mappings
zbMATH Keywords: KKM-type theorems
zbMATH Keywords: coincidence theorems
MaRDI profile type: Publication
full work available at URL: j.na.2008.11.055
OpenAlex ID: W2079597611
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DOI: 10.1016/J.NA.2008.11.055
description: scientific article; zbMATH DE number 5569417