Uniformly bounded composition operators in the Banach space of bounded \((p, k)\)-variation in the sense of Riesz-Popoviciu
type: Item
instance of: scholarly article
title: Uniformly bounded composition operators in the Banach space of bounded \((p, k)\)-variation in the sense of Riesz-Popoviciu
zbMATH Open document ID: 1296.47054
author: Francy Armao
author: D. Głazowska
author: Sergio Rivas
author: Jessica Rojas
published in: Central European Journal of Mathematics
publication date: 2013-02-18T00:00:00Z
review text: Let \([a,b]\) be a compact interval in \(\mathbb R\) and denote by \(RV_{(p,k)}[a,b]\) the vector space of all the functions \(u:[a,b] \to \mathbb R\) such that \(\hat{V}_{(p,k)}^R(u)<+\infty\), that is, such functions which have bounded \((p,k)\) variation in the sense of Riesz-Popoviciu, endowed with the norm \[ \|u\|_{(p,k)}^R=|u(a)|+|u'(a)|+\ldots+|u^{(k-1)}(a)|+(\hat{V}_{(p,k)}^R(u))^{\frac{1}{p}}. \] Let \(p>1\), \(k \geq 2\) a positive integer and let \(f:[a,b]\times\mathbb R \to \mathbb R\) be continuous with respect to the second variable. The main result of this article states that, if the nonautonomous superposition operator \(F\), generated by \(f\), maps the space \(RV_{(p,k)}[a,b]\) into itself and satisfies the inequality \[ \|F(u)-F(v)\|_{(p,k)}^R\leq\gamma (\|u-v\|_{(p,k)}^R),\quad u,v\in RV_{(p,k)}[a,b], \] for some function \(\gamma:[0,+ \infty) \to [0,+ \infty)\), then \(F\) possesses the Matkowski property, that is, there exist functions \(\alpha, \beta \in RV_{(p,k)}[a,b]\) such that \[ f(t,x)= \alpha(t)x+ \beta(t),\; t \in [a,b],\; x \in \mathbb R. \]
reviewed by: Dariusz Bugajewski
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 26A45
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 6137071
zbMATH Keywords: Nemytskij (composition
zbMATH Keywords: superposition) operator
zbMATH Keywords: uniformly bounded mapping
zbMATH Keywords: uniformly continuous mapping
zbMATH Keywords: de la Vallée Poussin second-variation
zbMATH Keywords: Popoviciu \(k\)-th variation
MaRDI profile type: Publication
full work available at URL: s11533-012-0051-5
OpenAlex ID: W2043341922
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DOI: 10.2478/S11533-012-0051-5
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description: scientific article; zbMATH DE number 6137071