Continuous selection theorems for Fan-Browder mappings in topological spaces and their applications
type: Item
instance of: scholarly article
title: Continuous selection theorems for Fan-Browder mappings in topological spaces and their applications
zbMATH Open document ID: 1148.54007
author: Ming-Ge Yang
author: Lei Deng
published in: Applied Mathematics and Mechanics. (English Edition)
publication date: 2008-09-17T00:00:00Z
review text: If \(X\) is a nonempty set and \(Y\) is a topological space, a set-valued mapping \(W:X\multimap Y\) is said to be a generalized \(R\)-\(KKM\)-mapping if for each \(A\in\langle D\rangle\) with the cardinality \(|A|=n+1\) there exists a continuous function \(\phi_A:\Delta_n\rightarrow Y\) such that \(J\in\langle A\rangle\) implies \(\phi_A(\Delta_J)\subset W(J)\), where \(\Delta_J\) is the standard subsimplex of \(\Delta_n\) corresponding to \(J\). Let \(X\) and \(Y\) be topological spaces and \(D\) be a nonempty subset of \(Y\). A set-valued mapping \(T:X\multimap Y\) is called a \(\Phi\)-mapping if there exists a generalized \(R\)-\(KKM\)-mapping \(W:D\multimap Y\) and a set-valued mapping \(S:X\multimap D\) having the local intersection property such that \(W(M)\subset T(x)\), for each \(x\in X\) and \(M\in \langle S(x)\rangle\). The main result of the paper is the following continuous selection theorem: Let \(X\) be a normal space, \(Y\) a topological space, \(D\) a nonempty subset of \(Y\). A \(\Phi\)-mapping \(T:X\multimap Y\) has a continuous selection whenever there exists a compact subset \(K\) of \(X\) and a finite subset \(M\) of \(D\) such that \(X\setminus K\subset \bigcup\limits_{y\in M} \text{int}\, S^-(y)\),where \(S:X\multimap D\) is the companion mapping of \(T\) in the definition above. As applications, fixed point theorems, coincidence theorems and a nonempty intersection theorem are given.
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5344601
zbMATH Keywords: Fan-Browder mapping
zbMATH Keywords: continuous selection
zbMATH Keywords: fixed point
zbMATH Keywords: coincidence point
reviewed by: Mircea Balaj
MaRDI profile type: Publication
full work available at URL: s10483-006-0409-z
OpenAlex ID: W2026013258
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DOI: 10.1007/S10483-006-0409-Z
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description: scientific article; zbMATH DE number 5344601