Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations
type: Item
instance of: scholarly article
title: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations
zbMATH Open document ID: 1193.47045
author: Ciprian Preda
author: Petre Preda
author: Aurelian Craciunescu
published in: Journal of Functional Analysis
publication date: 2010-01-11T00:00:00Z
review text: Let \(X\) be a Banach space, \(\Theta\) a metric space, and \((E(\mathbb R_+, X),F(\mathbb R_+, X))\) a pair of vector-valued Schäffer function spaces. Write \(\alpha(t,E)=\inf\{\alpha>0\mid \int^t_0|f(t)|\,ds\leq \alpha\|f\|_{E(\mathbb R_+,\mathbb R)}\), \(\forall f\in E(\mathbb R_+,\mathbb R)\}\), and \(\beta(t,F)=\|\chi_{[0,t]}\|_{F(\mathbb R_+, \mathbb R)}\). The autors show that if \((E(\mathbb R_+, X),F(\mathbb R_+, X))\) is uniformly admissible to an exponentially bounded, strongly continuous cocycle \(\{\pmb{\Phi}(\theta,t)\}_{\theta\in\Theta, t\geq 0}\) (over a semiflow \(\{\sigma(\theta,t)\}_{\theta\in\Theta, t\geq 0}\)) and \(\lim_{t\to\infty}\alpha(t,E)\beta(t,F)=\infty\), then there exist a family of projectors \(\{P(\theta)\}_\Theta\) and positive constants \(N_1\), \(N_2\), \(\nu_1\), \(\nu_2\) such that, for \(t\geq 0\), \(\theta\in\Theta\), \[ \|\pmb{\Phi}(\theta,t)x\|\leq N_1e^{-\nu_1 t}\|x\| \quad (x\in \operatorname{Im} P(\theta)); \qquad \|\pmb{\Phi}(\theta,t)x\|\geq N_2e^{\nu_2 t}\|x\| \quad (x\in \operatorname{Ker} P(\theta)). \]
reviewed by: Ti-Jun Xiao
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 5655695
zbMATH Keywords: (nonlinear) (semi)flows
zbMATH Keywords: exponentially bounded
zbMATH Keywords: strongly continuous cocycles
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.09.002
OpenAlex ID: W1985663844
cites work: Ordinary differential equations. An introduction to nonlinear analysis. Transl. from the German by Gerhard Metzen
cites work: Q4258125
cites work: Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces
cites work: Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces
cites work: Q4040820
cites work: Dichotomies in stability theory
cites work: Q4774403
cites work: One-Parameter Semigroups for Linear Evolution Equations
cites work: Existence and robustness of exponential dichotomy of linear skew-product semiflows over semiflows
cites work: Evolution semigroups, translation algebras, and exponential dichotomy of cocycles
cites work: Q5599358
cites work: One-parameter semigroups of positive operators
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Robustness of exponential dichotomies in infinite-dimensional dynamical systems
cites work: Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations
cites work: Q5291005
cites work: \((L^p,L^q)\)-admissibility and exponential dichotomy of evolutionary processes on the half-line
cites work: Schäffer spaces and uniform exponential stability of linear skew-product semiflows
cites work: Schäffer spaces and exponential dichotomy for evolutionary processes
cites work: Navier-Stokes Equations on Thin 3D Domains. I: Global Attractors and Global Regularity of Solutions
cites work: Existence of dichotomies and invariant splittings for linear differential systems. II
cites work: Dichotomies for linear evolutionary equations in Banach spaces
cites work: Dynamics of evolutionary equations
cites work: Infinite-dimensional dynamical systems in mechanics and physics.
cites work: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line
DOI: 10.1016/J.JFA.2009.09.002
description: scientific article; zbMATH DE number 5655695