Metric fixed point theory on hyperconvex spaces: recent progress
type: Item
instance of: scholarly article
title: Metric fixed point theory on hyperconvex spaces: recent progress
zbMATH Open document ID: 1273.54052
author: Rafa Espínola
author: Pepa Lorenzo
published in: Arabian Journal of Mathematics
publication date: 2013-01-02T00:00:00Z
review text: A metric space \(M\) is said to be hyperconvex if given any family \(\{x_{\alpha}\}\) of points of \(M\) and any family \(\{r_{\alpha}\}\) of real numbers satisfying \( d(x_{\alpha}, x_{\beta}) \leq r_{\alpha}+r_{\beta}\), then \(\bigcap_{\alpha} B(x_{\alpha},r_{\alpha}) \neq \varnothing\). In this survey paper, the authors review the development of metric fixed point theory on hyperconvex metric spaces. In Section 4, they discuss the problem of characterizing proximal nonexpansive retracts of hyperconvex spaces and its connections to several problems in best approximation and best proximity pairs. In Section 5, recent developments on \(\mathbb{R}\) trees and metric fixed point theory are treated. In the last section, some recent advances on the theory of extensions of Hölder maps and their relation to extensions of uniformly continuous mappings under \(\chi_0\) hyperconvex conditions are presented. Some new results on the extension of compact mappings are also given.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54-02
zbMATH DE Number: 6121002
zbMATH Keywords: hyperconvex metric spaces
zbMATH Keywords: proximal nonexpansive retracts
zbMATH Keywords: best approximation
zbMATH Keywords: best proximity pairs
MaRDI profile type: Publication
full work available at URL: s40065-012-0044-z
OpenAlex ID: W2143913670
Wikidata QID: Q59289472
cites work: A selection theorem in metric trees
cites work: Q5442466
cites work: Q4441888
cites work: A best approximation theorem in hyperconvex metric spaces
cites work: Best approximation, coincidence and fixed point theorems for set-valued maps in \(\mathbb R\)-trees
cites work: Coincidence point, best approximation, and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces
cites work: Q3515651
cites work: Extension of uniformly continuous transformations and hyperconvex metric spaces
cites work: Q3799313
cites work: Markov chains, Riesz transforms and Lipschitz maps
cites work: On fixed point theorems of Leray–Schauder type
cites work: On some properties of hyperconvex spaces
cites work: Q4720067
cites work: Measure of nonhyperconvexity and fixed-point theorems
cites work: Generalized KKM theorems on hyperconvex metric spaces and some applications
cites work: Fixed points of uniformly Lipschitzian mappings
cites work: Q3442330
cites work: Gated sets in metric spaces
cites work: Q4356024
cites work: Q5440763
cites work: Introduction to hyperconvex spaces
cites work: Fixed point theorems in \(\mathbb R\)-trees with applications to graph theory
cites work: Nonexpansive retractions in hyperconvex spaces
cites work: Extension of compact mappings and \(\aleph_0\)-hyperconvexity
cites work: Fixed points, selections and common fixed points for nonexpansive-type mappings
cites work: Q3998720
cites work: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant
cites work: Q3322631
cites work: On some covering and intersection properties in Minkowski spaces
cites work: Lectures on analysis on metric spaces
cites work: Nonsmooth calculus
cites work: Injective envelopes of Banach spaces are rigidly attached
cites work: KKM and Ky Fan theorems in hyperconvex metric spaces
cites work: Sadovskii's fixed point theorem without convexity
cites work: On the fixed points of commuting nonexpansive maps in hyperconvex spaces
cites work: An Introduction to Metric Spaces and Fixed Point Theory
cites work: Fixed point and selection theorems in hyperconvex spaces
cites work: Some generalizations of Ky Fan’s best approximation theorem
cites work: Lambda-hyperconvexity in metric spaces
cites work: Q3842812
cites work: Fixed point theorems in CAT(0) spaces and \(\mathbb R\)-trees
cites work: Krasnoselskii's iteration process in hyperbolic space
cites work: Best Approximation in ℝ-Trees
cites work: Remarks on best approximation in R-trees
cites work: Q4360768
cites work: Proximinal Retracts and Best Proximity Pair Theorems
cites work: Q4772256
cites work: On the extension of Hölder maps with values in spaces of continuous functions
cites work: On the fixed point set of nonexpansive order preserving maps
cites work: Extension of compact operators
cites work: Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces
cites work: Fixed points, selections and best approximation for multivalued mappings in \(R\)-trees
cites work: A best approximation theorem for nonexpansive set-valued mappings in hyperconvex metric spaces
cites work: Fixed-edge theorem for graphs with loops
cites work: Best approximation of coincidence points in metric trees
cites work: Hyperconvexity and approximate fixed points
cites work: Invariant approximation for CAT(0) spaces
cites work: On nonlinear contraction semigroups in sup norm spaces
cites work: Hyperconvexity and Nonexpansive Multifunctions

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cites work: Existence of Fixed Points of Nonexpansive Mappings in Certain Banach Lattices

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cites work: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings

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cites work: The Introduction of Local Connectivity by Change of Topology

Metric fixed point theory on hyperconvex spaces: recent progress (DOI)
DOI: 10.1007/S40065-012-0044-Z

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