Dynamics of a class of ODEs more general than almost periodic
type: Item
instance of: scholarly article
title: Dynamics of a class of ODEs more general than almost periodic
zbMATH Open document ID: 1217.34101
author: Juan-Miguel Gracia
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-03-14T00:00:00Z
review text: Given a Banach space \(X\), the authors investigate the function space \[ {\mathcal F}:=\{f:{\mathbb R}\to X\mid f\text{ is uniformly continuous with \(f({\mathbb R})\) being precompact}\} \] and provide interesting applications to solutions of nonautonomous ODEs in Banach spaces with respect to solutions in \({\mathcal F}\), the saddle-point structure around them, as well as pullback attraction properties. First, it is established that \({\mathcal F}\) is a closed subset of the bounded uniformly continuous functions. Moreover, provided \(A(t):D\to X\) with dense domain \(D\) generates an evolution operator, it is shown that \({\mathcal F}\) is an admissible space. This means, under exponential dichotomy, equicontinuity and compactness assumption on \(\dot x=A(t)x\), that the inhomogeneous problem \(\dot x=A(t)x+f(t)\) has a unique solution in \({\mathcal F}\) for every perturbation \(f\in{\mathcal F}\). After an insertion on sequential sets for \({\mathcal F}\) and dichotomy roughness, the authors verify that the above admissibility property also holds for weakly nonlinear problems \[ \qquad\dot x=A(t)x+f(t,x,\epsilon),\tag{\(*\)} \] provided \(f\) is Lipschitz in \(x\) and satisfies certain compactness properties. Around the resulting solution \(x^\ast(\epsilon)\) in \({\mathcal F}\), there exists a saddle-point structure whose stable and unstable integral manifolds are constructed using a Lyapunov-Perron approach. Applications to coupled heat and beam equations are given. Finally, under the assumption of a uniformly asymptotically stable linear part \(\dot x=A(t)x\), it is shown that \(x^\ast(\epsilon)\) is the pullback attractor of the weakly nonlinear problem \((\ast)\). The reviewer points out that for continuously Fréchet-differentiable \(f\) an alternative persistence proof for the solution \(x^\ast(\epsilon)\) and its saddle-point structure on the basis of the implicit mapping theorem might be possible [cf. the reviewer, Commun. Pure Appl. Anal. 10, No.~3, …
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 5865543
zbMATH Keywords: periodic solution
zbMATH Keywords: almost periodic solution
zbMATH Keywords: almost automorphic solution
zbMATH Keywords: sequential set
zbMATH Keywords: stable manifold
zbMATH Keywords: unstable manifold
zbMATH Keywords: pullback attractor
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.na.2010.12.025
OpenAlex ID: W2043722451
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DOI: 10.1016/J.NA.2010.12.025
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description: scientific article; zbMATH DE number 5865543