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Linearization and Lemma of Newton for Operator functions

  • We study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of G[z]=0. The results represent local generalizations of the standard implicit or inverse function theorem and of Newton's Lemma, considering the order of approximation needed to obtain solutions of G[z]=0. The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of k−1 with the givenWe study the action of the nonlinear mapping G[z] between real or complex Banach spaces in the vicinity of a given curve with respect to possible linearization, emerging patterns of level sets, as well as existing solutions of G[z]=0. The results represent local generalizations of the standard implicit or inverse function theorem and of Newton's Lemma, considering the order of approximation needed to obtain solutions of G[z]=0. The main technical tool is given by Jordan chains with increasing rank, used to obtain an Ansatz, appropriate for transformation of the nonlinear system to its linear part. The family of linear mappings is restricted to the case of an isolated singularity. Geometrically, the Jordan chains define a generalized cone around the given curve, composed of approximate solutions of order 2k with k denoting the maximal rank of Jordan chains needed to ensure k-surjectivity of the linear family. Along these lines, the zero set of G[z] in the cone is calculated immediately, agreeing up to the order of k−1 with the given approximation. Hence, the results may also be interpreted as a version of Tougeron's implicit function theorem or Hensel's Lemma in Banach spaces, essentially restricted to the arc case of a single variable. Finally, by considering a left shift of the Jordan chains, the Ansatz can be modified in a systematic way to obtain a sequence of refined versions of linearization theorems and Newton Lemmas in Banach spaces.show moreshow less

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Metadaten
Author:Matthias StiefenhoferORCiD
URN:urn:nbn:de:bvb:859-20857
Place of publication:Kempten
Document Type:Preprint
Language:English
Date of first Publication:2023/09/27
Publishing Institution:Hochschule für angewandte Wissenschaften Kempten
Tag:Linearization, Jordan chain, triangular Operator function, Newton Lemma, Tougeron implicit function theorem, Hensel Lemma
Number of pages:36 Seiten
First Page:1
Last Page:36
Original Publication / Source:Stiefenhofer, Matthias (2023): Linearisierung und Lemma von Newton für Operatorfunktionen, arXiv.org, New York, Cornell University
https://arxiv.org/abs/2309.15548,
https://doi.org/10.48550/arXiv.2309.15548
Institutes:Fakultät Maschinenbau
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik
Open Access:open_access
Research focus:Eigenständige Forschungsfelder
Publication Lists:Stiefenhofer, Matthias
Publication reviewed:nicht begutachtet
Licence (German):Creative Commons - CC BY-NC-ND - Namensnennung - Nicht kommerziell - Keine Bearbeitungen 4.0 International
Release Date:2024/01/29
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