Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems
type: Item
instance of: scholarly article
title: Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems
zbMATH Open document ID: 1063.90062
DOI: 10.1023/A:1023656507786
author: D. Massart
published in: Journal of Optimization Theory and Applications
publication date: 2003-10-29T00:00:00Z
review text: The paper derives Fan-Knaster-Kuratowski-Mazurkiewicz type theorems and geometric properties of convex spaces, applying these to obtain coincidence theorems and fixed point theorems in convex spaces, with which they obtain existence results for a solution of generalized vector equilibrium problems. Some details follow. Let \(X\) be a convex space, i.e. a non-empty convex set in a vector space with any topology that induces the Euclidean topology on the convex hulls of its finite subsets. Let \(Y\) be a Hausdorff topological space. If \( S,T:X\rightarrow 2^{Y}\)are maps such that \(T(coN)\sqsubseteq S(N)\), for each \(N\in \langle X\rangle \) (where \(coN\) denotes the convex hull of \(N\) and \( \langle X\rangle \) denotes the class of all finite subsets of \(X\) ) then \(S\) is said to be {generalized KKM mapping w.r.t. }\({T}\). The map \(T\) is said to have the {KKM property} if \(S\) is a generalized KKM map w.r.t. \(T\) such that the family \(\{\overline{S(X)}:x\in X\}\) has the finite intersection property, where \(\overline{S(X)}\) denotes the closure of \(S\left( X\right) \). Denote by \(KKM(X,Y)\) the family of all maps having the KKM property. A map \(G:X\rightarrow 2^{Y}\) is {transfer closed} if for any \(x\in X\) and \(y\notin P(x)\), there exists \(\widetilde{x}\in X\) such that \(y\notin P(\widetilde{x})\) and it is {transfer open} if for any \(x\in X\) and \(y\in P(x)\), there exists \(\widetilde{x}\in X\) such that \( y\in intP(\widetilde{x}).\) The following theorem is proved: let \(X\) be a convex space, \(Y\) a topological space and \(T\in KKM(X,Y)\) be compact. Assume that \( G:X\rightarrow 2^{Y}\) is transfer closed and, for any \(N\in \langle X\rangle ,T(coN)\sqsubseteq G(N).\) Then, \(\overline{T(X)}\cap [ \bigcap \{G(x):x\in X\}]\neq \varnothing .\) This leads to results concerning geometric properties of convex spaces and to coincidence theorems such as the following. Let \(Y\) be a convex space, \(X\) a topological space and \( T\in KKM(Y,…
Mathematics Subject Classification ID: 90C47
Mathematics Subject Classification ID: 47N10
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 1997751
zbMATH Keywords: Fan-KKM type theorem
zbMATH Keywords: coincidence theorems
zbMATH Keywords: fixed-point theorems
zbMATH Keywords: generalized vector equilibrium problems
zbMATH Keywords: maximal pseudomonotone maps
MaRDI profile type: Publication
cites work: A generalization of Tychonoff's fixed point theorem
cites work: Minimax and fixed point theorems
cites work: A new minimax inequality on H-spaces with applications
cites work: SURFACES OF GENERAL TYPE WITH pg= 1 AND q = 0
cites work: Geometric Properties, Minimax Inequalities, and Fixed Point Theorems on Convex Spaces
cites work: Q4231657
cites work: Generalizations of the FKKM theorem and the Ky Fan minimax inequality, with applications to maximal elements, price equilibrium, and complementarity
cites work: Applications of a fixed point theorem in \(G\)-convex space
cites work: A further generalization of Yannelis-Prabhakar's continuous selection theorem and its applications
cites work: On the use of KKM multifunctions in fixed point theory and related topics
cites work: KKM property and fixed point theorems
cites work: A KKM type theorem and its applications
cites work: Q3324260
cites work: On the generalized quasi-variational inequality problems
cites work: Coincidence Theorem and Saddle Point Theorem
cites work: The existence of nonlinear inequalities
cites work: Generalized monotone bifunctions and equilibrium problems
cites work: Q4366516
cites work: Equilibrium problems and noncoercive variational inequalities
cites work: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities
cites work: Duality for equilibrium problems under generalized monotonicity
cites work: Generalized variational inequalities and its applications
cites work: Q4762212
cites work: Vector equilibrium problems with generalized monotone bifunctions
cites work: Vector variational inequalities and vector equilibria. Mathematical theories
cites work: From scalar to vector equilibrium problems in the quasimonotone case
cites work: On noncooperative vector equilibrium
cites work: Fixed-point theorems and equilibrium problems
cites work: A remark on vector-valued equilibria and generalized monotonicity
cites work: On the existence of equilibrium points of vector functions
cites work: An existence result for the generalized vector equilibrium problem
cites work: On generalized vector equilibrium problems.
cites work: Existence of solutions for generalized vector equilibrium problems
cites work: Existence of equilibria for multivalued mappings and its application to vectorial equilibria
cites work: Existence of equilibria for monotone multivalued mappings
cites work: Generalized vector equilibrium problems
cites work: A generalization of vectorial equilibria
cites work: Q4762236
full work available at URL: a:1023656507786
OpenAlex ID: W2136172341
description: scientific article; zbMATH DE number 1997751