Essential critical points in product manifolds
type: Item
instance of: scholarly article
title: Essential critical points in product manifolds
zbMATH Open document ID: 0835.58008
published in: Rocky Mountain Journal of Mathematics
publication date: 1995-10-18T00:00:00Z
review text: Let \(M\) be a Banach space and \(f : M \to \mathbb{R}\) a continuous function. A point \(u \in M\) is said to be essentially regular for \(f\), if, up to a local homeomorphism, \(f\) is a nonconstant affine function near \(u\). A point \(u \in M\) is said to be essentially critical, if it is not essentially regular. These notions can be easily extended to the case where \(M\) is a Banach manifold and has a topological nature. They were introduced by \textit{F. Wille} [Contemp. Math. 72, 253-268 (1988; Zbl 0659.58004)] where the existence of such essential critical points was established when \(M\) is finite-dimensional. The authors consider a \(C^2\)-Finsler manifold \(M\), a metrizable Banach manifold \(P\) and a continuous function \(f : M \times P \to \mathbb{R}\) which is locally Lipschitz with respect to the first variable. They impose a kind of Palais-Smale condition, which seems to be rather natural when \(P\) is compact. Then they extend some of Wille's results to this infinite- dimensional setting. Finally, they give an application to the existence of periodic solutions for an ordinary differential equation.
reviewed by: Marco Degiovanni
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 796915
zbMATH Keywords: critical point theory
zbMATH Keywords: periodic solutions
zbMATH Keywords: ordinary differential equation
author: Gilles Fournier
author: Lech Górniewicz
author: Tomasz Kaczynski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072277
OpenAlex ID: W1998048358
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DOI: 10.1216/RMJM/1181072277
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description: scientific article; zbMATH DE number 796915