The structure of the zero sets of nonlinear mappings near generic multiparameter eigenvalues
type: Item
instance of: scholarly article
title: The structure of the zero sets of nonlinear mappings near generic multiparameter eigenvalues
zbMATH Open document ID: 0832.47050
author: Bryan P. Rynne
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-02-26T00:00:00Z
review text: Let \(X\) and \(Y\) be real Banach spaces, let \(f: \mathbb{R}^p\times X\to Y\) be a \(C^\ell\) mapping, \(\ell\geq 2\), and let \(L(\lambda)= Df (\lambda, 0)\), for \(\lambda\in \mathbb{R}^p\). Suppose also that \(f(\lambda, 0) =0\), for all \(\lambda\), and at some point \(\lambda^0\), \(\dim N(L (\lambda^0))= n>0\), \(\text{codim } R(L( \lambda^0))= m>0\). The author examines the structure of the zero set of \(f\) in a neighbourhood of \((\lambda^0, 0)\), and gives conditions under which this zero set is diffeomorphic to the zero set of the lower order terms in the Taylor expansion of \(f\) near \((\lambda^0, 0)\). These results tell something about the structure of the set of non-trivial solutions bifurcating from zero when there are several parameters and the dimension of the null space of the linearization is greater than 1. Finally, the author considers the genericity of the conditions used.
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 811227
zbMATH Keywords: structure of the zero set
zbMATH Keywords: Taylor expansion
zbMATH Keywords: solutions bifurcating from zero
zbMATH Keywords: genericity
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1291
OpenAlex ID: W1977417223
DOI: 10.1006/JMAA.1995.1291
description: scientific article; zbMATH DE number 811227