Resolving singular nonlinear equations
type: Item
instance of: scholarly article
title: Resolving singular nonlinear equations
zbMATH Open document ID: 0652.65046
DOI: 10.1216/RMJ-1988-18-2-225
published in: Rocky Mountain Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The authors develop a unified theory for the solution \(x_ 0=(z_ 0,\lambda_ 0)\) of an operator equation \(G(z,\lambda)=0\) where \(\lambda\in R\) q, dim N(G'(x\({}_ 0))=m+q\), and dim N(G'(x\({}_ 0)\) \(*)=m\). This formulation includes multiple bifurcations for multi- parameter problems \((q>0\), \(m>1)\), as well as singular operator equations \((q=0\), \(m>0)\). The problem is unfolded by embedding the operator G into a family of operators \(F=F(x,c)\) such that \(F(x,0)=G(x)\) and that with a suitable linear operator L the augmented map \[ H(x,w_ 1,...,x_{m+q},c)=(F(x,c),F_ x(x,c)w_ i,\quad Lw_ i-a_ i,\quad i=1,...,m+q) \] has an invertible derivative at \((x_ 0,...)\). This map H may be used to define quadratically convergent Newton methods. The finite dimensional case is studied in detail and then some discretizations of H and their computation are considered. Various examples are included and there is some discussion of determining \(m+q\) when, as is typical in applications, m is not known a priori.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 4064381
zbMATH Keywords: Banach spaces
zbMATH Keywords: multiple bifurcations
zbMATH Keywords: multi-parameter problems
zbMATH Keywords: singular operator equations
zbMATH Keywords: Newton methods
author: Eugene L. Allgower
author: Klaus Böhmer
reviewed by: Werner C. Rheinboldt
MaRDI profile type: Publication
full work available at URL: rmj-1988-18-2-225
OpenAlex ID: W2045423562
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description: scientific article; zbMATH DE number 4064381