A note on reflexivity and nonconvexity
type: Item
instance of: scholarly article
title: A note on reflexivity and nonconvexity
zbMATH Open document ID: 1244.47047
author: Isabel Marrero
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-10-27T00:00:00Z
review text: The Eisenfeld-Lakshmikantham (E-L, for short) measure of nonconvexity of a bounded subset of \(X\) is defined by \[ \mu(A)= \sup_{x\in\overline{\text{conv}(A)}} \underset{a\in A}{}{\text{inf}}\,\| x-a\|= H(A,\overline{\text{conv}(A))}), \] where \(H(C, D)\) is the Hausdorff-Pompeiu distance between the bounded subsets \(C\) and \(D\) of \(X\) (see [\textit{J. Eisenfeld} and \textit{V. Lakshmikantham}, Yokohama Math. J. 24, 133--140 (1976; Zbl 0361.34051)]). In the present paper, the author proves the following theorem. Theorem 2.5. For a Banach space \(X\), the following statements are equivalent: (i) \(X\) is reflexive; (ii) the E-L measure of nonconvexity \(\mu\) in \(X\) satisfies the Cantor property (i.e., for every decreasing sequence \(\{A_n\}^\infty_{n=1}\) of nonempty closed and bounded subsets of \(X\) such that \(\lim_{n\to\infty}\mu(A_n)= 0\), the closed and bounded set \(A_\infty= \bigcap^\infty_{n=1} A_n\) is nonempty and convex). Using this characterization, some results in best approximation and fixed point theory for reflexive Banach space are generalized by removing convexity requirements.
reviewed by: Jarosław Górnicki
Mathematics Subject Classification ID: 47H08
Mathematics Subject Classification ID: 41A52
Mathematics Subject Classification ID: 47H10
zbMATH DE Number: 5964217
zbMATH Keywords: best approximation
zbMATH Keywords: Cantor property
zbMATH Keywords: condensing map
zbMATH Keywords: contractive map
zbMATH Keywords: fixed point
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: measure of nonconvexity
zbMATH Keywords: reflexivity
zbMATH Keywords: strict convexity
MaRDI profile type: Publication
full work available at URL: j.na.2011.07.011
OpenAlex ID: W1996454210
cites work: Q4134286
cites work: Q5801080
cites work: Q3883053
cites work: Reflexivity and the sup of linear functionals
cites work: Q3977684
cites work: Measures of noncircularity and fixed points of contractive multifunctions
cites work: Reflexivity and approximate fixed points
cites work: Weak Compactness in Banach Spaces
cites work: Q5775858
cites work: An Introduction to Banach Space Theory
cites work: Proximality and Chebyshev sets
cites work: A Degree Theory, Fixed Point Theorems, and Mapping Theorems for Multivalued Noncompact Mappings
cites work: Q5186224
cites work: Q3678243
cites work: A Fixed Point Theorem for Mappings which do not Increase Distances
DOI: 10.1016/J.NA.2011.07.011
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description: scientific article; zbMATH DE number 5964217