Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations
type: Item
instance of: scholarly article
title: Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations
zbMATH Open document ID: 1169.34031
published in: Journal of Differential Equations
publication date: 2009-03-20T00:00:00Z
review text: The paper deals with the second-order differential equation \[ x''=f(\theta_th,x,x'). \tag{1} \] Here, \((H,d)\) is a compact metric space and \(\theta=(\theta_t)_{t\in R}\) is a dynamical system on \(H\). \(H\) is supposed to be minimal with respect to \(\theta\), that is \(\theta\) has no proper closed invariant subset in \(H\). Further \(f(h,x,p)\) is continuous on \(H\times (a,b)\times\mathbb R)\), satisfies the Lipschitz condition in \((x,p)\) and is strictly increasing in \(x\). Moreover \(f\) fulfils a quadratic growth condition in \(p\). The authors provide descriptions of the structure and behavior of bounded solutions of (1). They also prove the stability of the structure of bounded solutions with respect to parameter perturbations in \(f\). The proofs use the theory of pullback attractors for cocycle dynamical systems.
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 37C20
zbMATH DE Number: 5530619
zbMATH Keywords: nonautonomous dynamical systems
zbMATH Keywords: structure of bounded solutions sets
zbMATH Keywords: second-order differential equations
author: Desheng Li
author: Jinqiao Duan
MaRDI profile type: Publication
full work available at URL: j.jde.2008.10.031
OpenAlex ID: W2023761300
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DOI: 10.1016/J.JDE.2008.10.031
description: scientific article; zbMATH DE number 5530619