Locally uniform Opial conditions.
type: Item
instance of: scholarly article
title: Locally uniform Opial conditions.
zbMATH Open document ID: 1036.46013
DOI: 10.1016/S0362-546X(03)00023-3
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-05-14T00:00:00Z
review text: The authors consider several refinements of Opial's condition [Bull. Am. Math. Soc. 73, 591--597 (1967; Zbl 0179.19902)] that have appeared in the literature. A Banach space \(X\) satisfies a locally uniform Opial condition if, for every \(c>0\) and every weakly null sequence \(\{x_n\}\) in \(X\) with \(\limsup_n\| x_n \| \geq 1 \), there exists \(r>0\) such that \(\limsup_n \| x+x_n \| \geq 1+r\) for every \(x\in X\) with \(\| x \| \geq c\). A similar definition with the limit superiors replaced by limit inferiors could be considered and in [\textit{P. K. Lin, K. K. Tan} and \textit{H. K. Xu}, Nonlinear Anal., Theory Methods Appl. 24, 929--946 (1995; Zbl 0865.47040)] it is asserted that the definitions would be equivalent. In the article under review, the authors provide an example to show that, in general, the two notions are distinct. The authors also note that, for Banach spaces with separable duals, the limit inferior locally uniform Opial condition is equivalent to the uniform Opial property defined by \textit{S. Prus} in [Nonlinear Anal., Theory Methods Appl. 18, 697--704 (1992; Zbl 0786.46023)], but, in general, these properties also differ.
reviewed by: B. Turett
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 1910234
zbMATH Keywords: Opial condition
author: Helga Fetter
author: Berta Gamboa de Buen
MaRDI profile type: Publication
cites work: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings
cites work: Q4136031
cites work: Banach spaces with the uniform opial property
cites work: Uniform property (K) and its related properties
full work available at URL: s0362-546x(03)00023-3
OpenAlex ID: W2092854737
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description: scientific article; zbMATH DE number 1910234