A global inversion theorem for functions with singular points
type: Item
instance of: scholarly article
title: A global inversion theorem for functions with singular points
zbMATH Open document ID: 1423.46106
author: Piotr Fijałkowski
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2017-11-14T00:00:00Z
review text: The article deals with some generalization of the classic Hadamard-Levi and Krasnosel'skii theorem about global homeomorphism of a smooth map \(T\) between Banach spaces \(X\) and \(Y\) whose derivatives \(T'(x)\) are invertible for all \(x_0 \in X\). More precisely, the author considers the case when the derivatives \(T'(x)\) are invertible everywhere besides \(x = x_0\), and for some \(n\), \(n \geq 1\), \[ T^{(k)}(x_0) = 0, \quad k = 1,\dots,2n, \] \[ \|f'(x)h\| \geq C_1\|x\|^{2n}\|h\|, \quad C_2\|x\|^{2n+1} \leq \|f(x)\| \leq C_3\|x\|^{2n+1} \quad x \in X,\ h \in X, \] where \(f(x) = ((2n+1)!)^{-1} T^{(2n+1)}\). The main result is the following: Theorem. The map \(T\) is a homeomorphism \(X\) onto \(Y\) provided that \[ \sup_{x \in X \setminus \{0\}} \|r'(x)\| \|x\|^{-2n} < C_1C_2C_3^{-2n/(2n+1)}, \] where \(r(x)\) is a Taylor reminder term of the map \(T\) at the point \(x_0\). At the end of the article, the author provides two applications of this result.
reviewed by: Peter P. Zabreĭko
Mathematics Subject Classification ID: 46T20
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 47G20
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47J07
zbMATH DE Number: 6807049
zbMATH Keywords: inversion
zbMATH Keywords: differential map
zbMATH Keywords: singular point
zbMATH Keywords: nonlinear equation
zbMATH Keywords: integro-differential operator
MaRDI profile type: Publication
full work available at URL: dcdsb.2018011
OpenAlex ID: W2765090697
cites work: Local inversion theorem for singular points
cites work: On a Domain of Invertibility of a Differentiable Mapping
cites work: On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory
cites work: Certainty, doubt, error: comments on the epistemological foundations of medieval Arabic science
cites work: Q3671563
cites work: On the Diffeomorphisms Between Banach and Hilbert Spaces
cites work: Mountain pass theorems and global homeomorphism theorems
cites work: Homeomorphisms Between Banach Spaces
cites work: Global inversion theorems and applications to differential equations
cites work: An Application of Hadamard-Levy's Theorem to a Scalar Initial Value Problem
cites work: Diffeomorphisms with Banach space domains
DOI: 10.3934/DCDSB.2018011
description: scientific article; zbMATH DE number 6807049