Differential equations with locally bounded delay
type: Item
instance of: scholarly article
title: Differential equations with locally bounded delay
zbMATH Open document ID: 1241.34072
author: Hans-Otto Walther
published in: Journal of Differential Equations
publication date: 2012-03-07T00:00:00Z
review text: The initial value problem with state-dependent delays \[ x'(t)=g(x_t), \quad x_0=\phi, \] is studied for a continuously differentiable map \(g: U\to \mathbb R,\) where \(U\) is a non-empty open subset of a Banach space. It is proved that the set \(M:=\{\phi\in U: \phi'(0)=g(\phi)\}\) is a continuously differentiable submanifold with codimension \(n\) and that the above initial value problem defines a continuous semiflow of continuously differentiable solution operators on \(M.\) It is also verified that the derivative of the solution operators are given by solutions of linear variational equations with segments tangent to \(M.\) Local stable and unstable manifolds at a stationary point of the semiflow on \(M\) are provided.
reviewed by: Sotiris K. Ntouyas
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 34K19
Mathematics Subject Classification ID: 58B99
Mathematics Subject Classification ID: 34K21
zbMATH DE Number: 6012777
zbMATH Keywords: functional differential equation
zbMATH Keywords: state-dependent delay
zbMATH Keywords: infinite delay
zbMATH Keywords: local invariant manifold
MaRDI profile type: Publication
full work available at URL: j.jde.2011.11.004
OpenAlex ID: W2058513789
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DOI: 10.1016/J.JDE.2011.11.004
description: scientific article; zbMATH DE number 6012777