A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations
type: Item
instance of: scholarly article
title: A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations
zbMATH Open document ID: 1190.34065
author: J. Blot
published in: Journal of Differential Equations
publication date: 2009-07-09T00:00:00Z
review text: The main purpose of the paper is to provide a simple and \(C_0\)-semigroup free approach to the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations (see, e.g., [\textit{W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander}, Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96. (Basel): Birkhäuser. (2001; Zbl 0978.34001)]). First, a new approach to the concept of reduced spectrum on \(R\) or on \(R^+\) is presented. As a result, a new relation between the reduced spectrum and the spectrum of the differentiation operator is established. Gelfand's theorem is extended as the key tool for proving Loomis' theorem. A version of Loomis' theorem with an ergodic condition is then derived and used to obtain properties of the differentiation operator on the half-line. In the application of these results to stability of evolution equations, among other things, well-posedness assumption can be omitted. Reviewer's remark: In nearly the whole paper the author claims that his results are true without the assumption that functions considered are uniformly continuous. However, it is said in the final remark that uniform continuity is still crucial.
reviewed by: Zoran Kadelburg
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34C27
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 5577116
zbMATH Keywords: Spectrum of a function
zbMATH Keywords: reduced spectrum
zbMATH Keywords: Arendt-Batty-Lyubich-Vu theorem
zbMATH Keywords: almost periodicity
zbMATH Keywords: almost automorphy
zbMATH Keywords: asymptotic stability
MaRDI profile type: Publication
full work available at URL: j.jde.2009.03.030
OpenAlex ID: W1531213803
cites work: Tauberian Theorems and Stability of One-Parameter Semigroups
cites work: Almost periodic solutions of first- and second-order Cauchy problems
cites work: Asymptotically Almost Periodic Solutions of Inhomogeneous Cauchy Problems on The Half-Line
cites work: Q2716079
cites work: Discrete spectrum and almost periodicity
cites work: Harmonic analysis and asymptotic behavior of solutions to the abstract Cauchy problem
cites work: Asymptotic behavior of solutions of systems of neutral and convolution equations
cites work: Relations between different types of spectra and spectral characterizations
cites work: Q4400389
cites work: Q3209125
cites work: Q3219278
cites work: Tauberian theorems and stability of solutions of the Cauchy problem
cites work: Local spectra and individual stability of uniformly bounded $C_0$-semigroups
cites work: Equality of two spectra arising in harmonic analysis and semigroup theory
cites work: A NEW APPROACH TO ALMOST PERIODICITY
cites work: Q3906660
cites work: Stability and almost periodicity of solutions of ill-posed abstract Cauchy problems
cites work: Almost automorphic solutions of evolution equations
cites work: Q3965074
cites work: Bounded Solutions of Parabolic Equations in Continuous Function Spaces
cites work: Circular spectrum and bounded solutions of periodic evolution equations
cites work: The spectral characterization of a class of almost periodic functions
cites work: Asymptotic stability of linear differential equations in Banach spaces
cites work: A spectral criterion for the almost periodicity of one-parameter semigroups
cites work: Katznelson-Tzafriri type theorems for individual solutions of evolution equations
cites work: A spectral countability condition for almost automorphy of solutions of differential equations
cites work: Evolution semigroups and sums of commuting operators: A new approach to the admissibility theory of function spaces
cites work: Massera's theorem for almost periodic solutions of functional differential equations
cites work: New spectral criteria for almost periodic solutions of evolution equations
cites work: Individual stability of \(C_0\)-semigroups with uniformly bounded local resolvent
cites work: Almost automorphic and almost periodic functions in abstract spaces
cites work: Stability and almost periodicity of trajectories of periodic processes
cites work: Q4352387
cites work: Q4276447
cites work: Investigations in Harmonic Analysis
cites work: Asymptotically almost periodic solutions of evolution equations in Banach spaces
cites work: Almost automorphic and almost periodic dynamics in skew-product semiflows
cites work: Q3671556
cites work: Q3249198
cites work: Almost Automorphic Functions on Groups
cites work: Q4676191
cites work: Q5531981
DOI: 10.1016/J.JDE.2009.03.030
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description: scientific article; zbMATH DE number 5577116