A substitute for the Sard-Smale theorem in the \(C^1\) case
type: Item
instance of: scholarly article
title: A substitute for the Sard-Smale theorem in the \(C^1\) case
zbMATH Open document ID: 0924.58001
DOI: 10.1007/BF02786938
author: Jacobo Pejsachowicz
author: P. J. Rabier
published in: Journal d'Analyse Mathématique
publication date: 1999-11-08T00:00:00Z
review text: The purpose of this paper is to find a Fredholm perturbation of a Fredholm mapping \(F\) of index \(\tau \in\mathbb{Z},\) which is small in the \(C^1\) sense, has 0 as a regular value and can be connected to \(F\) by a \(C^1\) Fredholm homotopy with compact zero set if \(F\) is proper. Let us observe that the Fredholm mapping \(F\) is only \(C^1,\) yet of arbitrary index. The Sard-Smale theorem yields the existence of arbitrary small perturbations of a Fredholm mapping of index \(\tau \in\mathbb{Z}\) and class \(C^{{\max(\tau,0)}+1}\) between separable Banach spaces that have 0 as a regular value. In this theorem, the smoothness hypotheses can be weakened if \(\tau \geq 0,\) but the authors prove in this paper that the approximation results remain valid irrespective of the index \(\tau\) when \(F\) only belongs to \(C^1\) and verifies properness-like conditions. The hypotheses of separable spaces is not needed either. The authors also prove that in Banach spaces the hypotheses about \(C^1\) norm can be dropped. In order to state these results, the authors point out that all the arguments go over to the setting of Banach manifolds modeled on spaces equipped with a norm of \(C^1\) class away from the origin.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 58B15
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 46E25
zbMATH DE Number: 1274832
zbMATH Keywords: Sard-Smale theorem
zbMATH Keywords: Fredholm perturbation
zbMATH Keywords: Banach manifold
zbMATH Keywords: approximation of homotopies
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02786938
OpenAlex ID: W2008479398
description: scientific article; zbMATH DE number 1274832