A concept of convergence in geodesic spaces
type: Item
instance of: scholarly article
title: A concept of convergence in geodesic spaces
zbMATH Open document ID: 1145.54041
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2008-06-10T00:00:00Z
review text: In the present paper, a \textit{CAT(0) space} is a geodesic space for which each geodesic triangle is at least as `thin' as its comparison triangle in the Euclidean plane. A notion of convergence introduced independently by \textit{T.-C. Lim} [Proc. Am. Math. Soc. 60, 179--182 (1976; Zbl 0346.47046)] and \textit{T. Kuczumow} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 32, 79--88 (1978; Zbl 0463.47035)] is shown in CAT(0) spaces to be very similar to the usual weak convergence in Banach spaces. In particular, many Banach space results involving weak convergence have precise analogues in this setting. The paper ends with several open questions.
reviewed by: In-Sook Kim
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 5285994
zbMATH Keywords: CAT(0) space
zbMATH Keywords: \(\varDelta \)-convergence
zbMATH Keywords: weak convergence
zbMATH Keywords: fixed point
author: W. A. Kirk
author: B. Panyanak
MaRDI profile type: Publication
full work available at URL: j.na.2007.04.011
OpenAlex ID: W2155802942
cites work: Q4720067
cites work: A course in metric geometry
cites work: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces
cites work: Q3442330
cites work: Fixed points of uniformly Lipschitzian mappings
cites work: The fixed point property for mappings admitting a center
cites work: Fixed point theory for almost convex functions
cites work: Q3998720
cites work: Q3322631
cites work: La propriété du point fixe dans les espaces de Banach avec base inconditionelle. (The fixed point properties in Banach spaces with unconditional bases)
cites work: Nonexpansive mappings in metric and Banach spaces
cites work: Q4462019
cites work: Q3353650
cites work: An ultrafilter approach to locally almost nonexpansive maps
cites work: Q3913946
cites work: Remarks on Some Fixed Point Theorems
cites work: Degree theory for local condensing maps
cites work: Fixed points of condensing functions
cites work: Nonexpansive iterations in hyperbolic spaces
cites work: Locally almost nonexpansive mappings
cites work: Analogs of weak convergence in a special metric space
DOI: 10.1016/J.NA.2007.04.011
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description: scientific article; zbMATH DE number 5285994