A generalization of the implicit function theorem for mappings from \({\mathcal R}^{n+1}\) into \({\mathcal R}^ n\) and its applications
type: Item
instance of: scholarly article
title: A generalization of the implicit function theorem for mappings from \({\mathcal R}^{n+1}\) into \({\mathcal R}^ n\) and its applications
zbMATH Open document ID: 0534.58007
DOI: 10.1016/0022-1236(84)90085-5
published in: Journal of Functional Analysis
publication date: 1984-01-01T00:00:00Z
review text: This paper is devoted to the proof of a generalization of the implicit function theorem to mappings from \(R^{n+1}\) into \(R^ n\). Under some appropriate nondegeneracy condition it is shown that the zero set of such a mapping around the origin (say) is the union of a finite number of curves. An upper bound for this number is derived from Bezout's theorem. In comparison with earlier results in the same direction by various authors, the main improvement is the observation that the curves actually have one more degree of regularity than expected at the origin. in the applications to one-parameter bifurcation problems, this improvement is important because it allows one to assumme one fewer degree of regularity of the data on the one hand and leads to a homogeneous exposition of all the cases on the other hand. Further, the ''almost'' constructive proofs have led to new algorithms of computation of the bifurcated branches which are described and studied elsewhere.
Mathematics Subject Classification ID: 58C15
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 3848258
zbMATH Keywords: Morse lemma
zbMATH Keywords: nondegeneracy condition
zbMATH Keywords: homogeneous polynomial mapping
author: P. J. Rabier
MaRDI profile type: Publication
full work available at URL: 0022-1236(84)90085-5
OpenAlex ID: W1998301664
cites work: Applications of the blowing-up construction and algebraic geometry to bifurcation problems
cites work: Bifurcation from simple eigenvalues
cites work: Bifurcation Theory in Real Banach Space
cites work: Q3274351
cites work: Loss of stability and bifurcation at a double eigenvalue
cites work: On the local structure of the zero-set of a Banach space valued mapping
cites work: Qualitative methods in bifurcation theory
cites work: A Synthetic Study of One-Parameter Nonlinear Problems
cites work: Local structure of the zero-sets of differentiable mappings and application to bifurcation theory.
description: scientific article; zbMATH DE number 3848258