Center manifolds and contractions on a scale of Banach spaces
type: Item
instance of: scholarly article
title: Center manifolds and contractions on a scale of Banach spaces
zbMATH Open document ID: 0621.47050
DOI: 10.1016/0022-1236(87)90086-3
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: The paper contains a new proof of the existence of a \(C^ k\)-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vectorfield of class \(C^ k\). The main idea is that (after an appropriate globalization of the problem) such center manifold contains all solutions which have a certain maximal exponential growth at infinity. This leads to a fixed point problem on a scale of Banach spaces, indexed by the exponent; the difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on this scale of Banach spaces, such that a direct application of the implicit function theorem is not possible. The proof is then based on an abstract theorem which gives the differentiable dependence on parameters of the fixed point of uniform contractions on embedded Banach spaces and under conditions satisfied in particular by the center manifold problem.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 4007068
zbMATH Keywords: existence of a \(C^ k\)-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vectorfield of class \(C^ k\)
zbMATH Keywords: fixed point problem on a scale of Banach spaces
zbMATH Keywords: fixed point of uniform contractions on embedded Banach spaces
zbMATH Keywords: center manifold problem
author: A. Vanderbauwhede
author: Stephan A. van Gils
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90086-3
OpenAlex ID: W1998618993
cites work: Applications of centre manifold theory
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cites work: Q3860165
cites work: On the Application of Integral Manifolds to Hopf Bifureaction
cites work: Q3854019
cites work: Properties of Center Manifolds
cites work: Q3833289
description: scientific article; zbMATH DE number 4007068