Uniformly continuous composition operators in the space of bounded \(\varphi \)-variation functions
type: Item
instance of: scholarly article
title: Uniformly continuous composition operators in the space of bounded \(\varphi \)-variation functions
zbMATH Open document ID: 1225.47078
author: Janusz Matkowski
author: José Atilio Guerrero
author: Hugo Leiva
author: N. Merentes
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-02-12T00:00:00Z
review text: The authors write: ``We prove that, under some general assumptions, a generator of any uniformly continuous Nemytskii operator, mapping a subset of [the] space of bounded variation functions in the sense of Wiener into another space of this type, must be an affine function.'' An important ingredient in the proof is a result about left regularization by \textit{V. V. Chistyakov} [J. Math. Sci., 110, No.~2, 2455--2466 (2002; Zbl 1033.26013)]. The authors show that the left regularization \(h^-\) of the generating function \(h\) satisfies the Jensen functional equation and by modifying the standard argument [see \textit{M. Kuczma}, ``An introduction to the theory of functional equations and inequalities'' (Prace Naukowe Uniwersytetu Ślaskiego w Katowicach 489; Warszawa-Kraków-Katowice: Państwowe Wydawnictwo Naukowe) (1985; Zbl 0555.39004)], they obtain that \(h^-\) is an affine function. Reviewer's remarks. In the statement of their Theorem 3.1, the authors should have spelt out that the space \(Y\) is not only a normed space, but a Banach space. Also, the proof of the theorem seems to have a gap, consequently their Remark 3.5 appears to be incorrect. Fortunately, the gap can easily be filled if, instead of using the functions \(\eta_{\alpha,\beta}\), more general piecewise linear functions are used.
reviewed by: Héctor N. Salas
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 26A45
zbMATH DE Number: 5669299
zbMATH Keywords: \(\varphi\)-variation in the sense of Wiener
zbMATH Keywords: uniformly continuous operator
zbMATH Keywords: regularisation
zbMATH Keywords: composition
zbMATH Keywords: Jensen equation
MaRDI profile type: Publication
full work available at URL: j.na.2009.11.051
OpenAlex ID: W2026330980
cites work: Q3229647
cites work: Q5801558
cites work: Q3847669
cites work: Uniformly continuous superposition operators in the space of bounded variation functions
cites work: Q5848315
cites work: Q5765474
cites work: On generalized variations (I)
cites work: Composing Functions of Bounded ϕ-Variation
cites work: On some propertis of functions of generalized variation
cites work: Mappings of generalized variation and composition operators
cites work: On a Characterization of Lipschitzian Operators of Substitution in the SpaceBV{a, b)
cites work: Q3218572
cites work: Some elementary examples in nonlinear operator theory
DOI: 10.1016/J.NA.2009.11.051
description: scientific article; zbMATH DE number 5669299