Partial linearization for noninvertible mappings
type: Item
instance of: scholarly article
title: Partial linearization for noninvertible mappings
zbMATH Open document ID: 0813.47075
DOI: 10.1007/BF00991895
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1994-11-20T00:00:00Z
review text: The authors prove a Hartman-Grobman result for noninvertible mappings in a rather general situation: Let \({\mathcal U}\), \(\mathcal V\), \(\mathcal W\) be Banach spaces and \(F: {\mathcal U}\to {\mathcal U}\), \(G: {\mathcal V}\to {\mathcal V}\), \(H: {\mathcal W}\to {\mathcal W}\) bounded linear operators such that there is an \(a\in (0,1)\) with \(\sigma(F)\subset \{\lambda\in \mathbb{C}\mid|\lambda |< a\}\), \(\sigma(G)\subset \{\lambda\in \mathbb{C}\mid a< | \lambda|< 1\}\), and \(\sigma(H)\subset \{\lambda\in \mathbb{C}\mid |\lambda|> 1\}\). The Banach spaces are assumed to possess a smooth norm in the sense that there exists a real-valued \(C^ 1\)- function \(\lambda\) with bounded and uniformly Lipschitzian derivative which vanishes outside the unit ball and equals 1 on some ball around zero. They consider nonlinear operators on a neighbourhood \(\mathcal N\) of zero in \({\mathcal E}:= {\mathcal U}\times {\mathcal V}\times {\mathcal W}\), \(U: {\mathcal N}\to {\mathcal U}\), \(V: {\mathcal N}\to {\mathcal V}\), \(W: {\mathcal N}\to {\mathcal W}\) of the form \(U(u,v,w)= Fu+ f(u,v,w)\), \(V(u,v,w)= Gv+ g(u,v,w)\), \(W(u,v,w)= Hw+ h(u,v,w)\) where \(f\), \(g\), \(h\) are assumed to be partially differentiable with respect to \(u\), \(v\), \(w\) with partial derivatives which are bounded and uniformly Lipschitzian in \(u\), \(v\), \(w\). Moreover, \(f\), \(g\), \(h\) are assumed to be flat at zero (i.e., \(f\), \(g\), \(h\) and their first partial derivatives with respect to \(u\), \(v\), \(w\) take the value zero at zero). It is conjectured that under these conditions there is an open neighbourhood \({\mathcal N}_ 0\) of zero in \(\mathcal E\) and a continuous function \(f_ 0: {\mathcal N}_ 0\to {\mathcal U}\) with \(f_ 0(0,0,0)= 0\) such that \((U,V,W)\) is locally conjugate to \((F+ f_ 0,G,H)\) (i.e., there is a homeomorphism of neighbourhoods of zero in \(\mathcal E\) which carries \((U,V,W)\) to \((F+ f_ 0,G,H)\)). (For example, the classical Hartman-Gro…
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 58C40
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 690937
zbMATH Keywords: Hartman-Grobman linearization
zbMATH Keywords: local conjugacy
zbMATH Keywords: Hartman-Grobman result
zbMATH Keywords: noninvertible mappings
zbMATH Keywords: uniformly Lipschitzian derivative
zbMATH Keywords: nonlinear operators
author: Bernd Aulbach
author: Barnabas M. Garay
reviewed by: Christian Fenske
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 690937