Continuity and Fréchet-differentiability of Nemytskij operators in Hölder spaces
type: Item
instance of: scholarly article
title: Continuity and Fréchet-differentiability of Nemytskij operators in Hölder spaces
zbMATH Open document ID: 0765.47022
DOI: 10.1007/BF01303062
author: S. Singh
published in: Monatshefte für Mathematik
publication date: 1993-05-12T00:00:00Z
full work available at URL: https://eudml.org/doc/178556
review text: The paper is a continuation of \textit{M. Goebel} [Glasg. Math. J. 33, No. 1, 1-5 (1991; Zbl 0724.47041)] and deals with Nemytskij operators (superposition operators), \((Fy)(t)=f(t,y(t))\), which are generated by a function \(f: [a,b]\times\mathbb{R}^ n\to\mathbb{R}\). There are given sufficient conditions for \(f\), such that \(F\) is a bounded, continuous, locally Lipschitz continuous and continuously Fréchet-differentiable, respectively, map between the Hölder spaces \(H^ \alpha([a,b],\mathbb{R}^ n)\) and \(H^ \alpha[a,b]\), \(0<\alpha\leq 1\). The presented criteria require less strong assumptions than those so far known from the literature; all proofs are straightforward.
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 58C07
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 58C20
zbMATH DE Number: 164939
zbMATH Keywords: superposition operators
zbMATH Keywords: Nemytskij operators
zbMATH Keywords: continuously Fréchet- differentiable
zbMATH Keywords: Hölder spaces
reviewed by: Manfred Goebel
MaRDI profile type: Publication
cites work: On the differentiability of the superposition operator in hölder and Sobolev spaces
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description: scientific article; zbMATH DE number 164939