A general coincidence theory for set-valued maps
type: Item
instance of: scholarly article
title: A general coincidence theory for set-valued maps
zbMATH Open document ID: 0938.47036
DOI: 10.4171/ZAA/907
author: Donal O'Regan
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2000-06-09T00:00:00Z
full work available at URL: https://eudml.org/doc/48435
review text: The author derives coincidence theorems for multivalued maps admitting continuous selections by translating this problem into a fixed point problem for multivalued maps with acyclic values. His main fixed point result reads as follows: Let \(E\) be a Banach space and \(Q\ni 0\) a closed convex subset of \(E\). Choose a retraction \(r:E\to Q\) and let \(G\) be a map which assigns to each \(x\in Q\) a nonempty closed acyclic subset of \(E\). Assume that \(G(Q)\) is bounded and that \(G\circ r\) is condensing with closed graph. Assume moreover that for each sequence \((x_n,\lambda_n)\) in \(\partial Q\times[0,1]\) which converges to \((x,\lambda)\in\partial Q\times[0,1)\) with \(x\in\lambda G(x)\) there is an \(n_0\) such that \(\lambda_nG(x_n)\subset Q\) whenever \( n\geq n_0\). The conclusion is that \(G\) has a fixed point in \(Q\).
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 1356404
zbMATH Keywords: acyclic map
zbMATH Keywords: coincidence
zbMATH Keywords: multivalued maps
zbMATH Keywords: continuous selections
zbMATH Keywords: fixed point problem for multivalued maps with acyclic values
reviewed by: Christian Fenske
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1356404