Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions.
type: Item
instance of: scholarly article
title: Fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions.
zbMATH Open document ID: 1056.47039
author: D. Massart
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
full work available at URL: https://idus.us.es/xmlui/handle/11441/45290
review text: The authors prove a fixed point theorem for a nonexpansive (compact convex-valued) multimap satisfying the inwardness condition on a bounded closed convex subset of a Banach space whose characteristic of noncompact convexity is less than~1. Reviewer's remarks: Some conditions of the theorem seem to be superfluous: (1) the bounded range of a multimap is the consequence of the boundedness of the domain and nonexpansivity of a multimap; (2) \(1-\chi\)-contractivity of a multimap also follows from its nonexpansivity. (The authors treat this as an open problem but it is the corollary of Proposition 2.2.2 of the book of \textit{M. Kamenskii}, the reviewer and \textit{P. Zecca} [``Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces'' (de Gruyter Series in Nonlinear Analysis and Applications 7, Berlin) (2001; Zbl 0988.34001)].
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 2056498
zbMATH Keywords: fixed point
zbMATH Keywords: multivalued map
zbMATH Keywords: multimap
zbMATH Keywords: nonexpansive map
zbMATH Keywords: inwardness condition
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: characteristic of noncompact convexity
zbMATH Keywords: Opial condition
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
OpenAlex ID: W1986862007
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DOI: 10.1016/J.JMAA.2003.10.019
description: scientific article; zbMATH DE number 2056498