An overview on the Prus-Szczepanik condition
type: Item
instance of: scholarly article
title: An overview on the Prus-Szczepanik condition
zbMATH Open document ID: 1359.46011
DOI: 10.1186/1687-1812-2014-37
author: Enrique Llorens-Fuster
author: Eva M. Mazcuñán-Navarro
author: Omar Muñiz-Pérez
author: Carlos A. Hernández-Linares
published in: Fixed Point Theory and Applications
publication date: 2015-03-20T00:00:00Z
review text: Let \(N(X)\) be the family of all weakly null sequences in \(S(X)\), and \(M(X)\) a certain subclass. Let \[ d(\varepsilon,x)= \inf_{(y_m)\in N(X)} \limsup_{m\to\infty} \| x+\varepsilon y_m\|-\| x\|, \] \[ b(\varepsilon,x)= \sup_{(y_m)\in N(X)} \liminf_{ m\to\infty} \| x+\varepsilon y_m\|-\| x\| \] and \[ b_1(\varepsilon,x)= \sup_{(y_m)\in M(X)} \liminf_{ m\to\infty} \| x+\varepsilon y_m\|-\| x\|. \] A real Banach space without the Schur property is defined to be nearly uniformly noncreasy if for every \(\varepsilon>0\) there exists \(t>0\) such that for every \(x\) in the unit sphere \(S(X)\) it is true that \(d(\varepsilon,x)\geq t\) or \(b(t,x)\leq\varepsilon t\). The space \(X\) satisfies the Prus-Szczepanik (PSz) condition if there exists \(\varepsilon> 0\) such that, for every \(x\) in \(S(X)\), either \(b_1(1,x)< 1-\varepsilon\) or \(d(1, x)>\varepsilon\). The results of the paper arise from a study of the relationship between the above properties, the PSz condition and other well-known geometrical properties of Banach spaces. The interesting results are too involved to be quoted here.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 6417859
zbMATH Keywords: fixed point property
zbMATH Keywords: \(E\)-convexity
zbMATH Keywords: (PS) condition
zbMATH Keywords: Opial modulus
zbMATH Keywords: uniformly noncreasy Banach spaces
Wikidata QID: Q59318529
reviewed by: Srinivasa Swaminathan
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1186/1687-1812-2014-37
OpenAlex ID: W2140574723
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description: scientific article; zbMATH DE number 6417859