Nonlinear perturbations of nonuniform exponential dichotomy on measure chains
type: Item
instance of: scholarly article
title: Nonlinear perturbations of nonuniform exponential dichotomy on measure chains
zbMATH Open document ID: 1243.37020
author: Meng Fan
author: Xiaoyuan Chang
author: Jimin Zhang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-12-08T00:00:00Z
review text: This paper deals with the following nonuniformly hyperbolic system on measure chains \[ x^\Delta=A(t)x, \] and its nonlinear perturbation \[ x^\Delta=A(t)x+f(t,x), \] where \(A\in C_{\text{rd}}(\mathbb{T},\mathcal{B}(\mathbb{X}))\) and \(f:\mathbb{T}\times\mathbb{X}\rightarrow\mathbb{X}\), while \(C_{\text{rd}}(\mathbb{T},\mathbb{X})\) denotes the set of rd-continuous functions \(g:\mathbb{T}\rightarrow\mathbb{X}\). By constructing the topological equivalence between the above systems, a generalized Grobman-Hartman theorem is established. In addition, stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy are constructed. It is also shown that the stable invariant manifolds are Lipschitz in the initial values if the nonlinear perturbation is a sufficiently small Lipschitz perturbation.
reviewed by: Feng Xie
Mathematics Subject Classification ID: 37C15
Mathematics Subject Classification ID: 37D10
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34N05
zbMATH DE Number: 5987799
zbMATH Keywords: topological equivalence
zbMATH Keywords: stable invariant manifolds
zbMATH Keywords: nonuniform exponential dichotomies
zbMATH Keywords: measure chains
MaRDI profile type: Publication
full work available at URL: j.na.2011.09.001
OpenAlex ID: W2024156002
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DOI: 10.1016/J.NA.2011.09.001
description: scientific article; zbMATH DE number 5987799