Discrete admissibility and exponential dichotomy for evolution families
type: Item
instance of: scholarly article
title: Discrete admissibility and exponential dichotomy for evolution families
zbMATH Open document ID: 1032.34048
author: Mihail Megan
author: Adina Luminiţa Sasu
author: Bogdan Sasu
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-03-05T00:00:00Z
review text: Here, the authors study connections between admissibility and uniform exponential dichotomy for discrete and continuous evolution families. In the discrete case, they show that, for an evolution family with uniform exponential growth, an exponential dichotomy is equivalent to the admissibility of the pair \((c_0(X),c_{00}(X))\), where \(X\) is a Banach space and the sequence spaces \(c_0(X)=\{s:{\mathbb Z}_+\rightarrow X: \lim_{n\to\infty}s(n)=0\}\), \(c_{00}(X)=\{s\in c_0(X): s(0)=0\}\) are endowed with the sup-norm. Having this result at hand, the authors are able to give a new proof of an admissibility theorem due to \textit{Nguyen Van Minh, F. Räbiger} and \textit{R. Schnaubelt} [Integral Equations Oper. Theory 32, 332--353 (1998; Zbl 0977.34056)]. This theorem states that, in a continuous setting, the uniform exponential dichotomy of a strongly continuous evolution family is equivalent to the admissibility of the pairs \((C_0(X),C_0(X))\) and \((C_0(X),C_{00}(X))\), respectively, with \(C_0(X)=\{s:{\mathbb R}_+\rightarrow X: \lim_{t\to\infty}s(t)=0\}\), \(C_{00}(X)=\{s\in C_0(X): s(0)=0\}\) equipped with the sup-norm.
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 1890743
zbMATH Keywords: evolution family
zbMATH Keywords: uniform exponential dichotomy
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: dcds.2003.9.383
OpenAlex ID: W2083389642
DOI: 10.3934/DCDS.2003.9.383
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description: scientific article; zbMATH DE number 1890743