Uniform dichotomy and exponential dichotomy of evolution families on the half-line
type: Item
instance of: scholarly article
title: Uniform dichotomy and exponential dichotomy of evolution families on the half-line
zbMATH Open document ID: 1113.34038
author: Bogdan Sasu
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: The author characterizes uniform and exponential dichotomies of evolution equations on half-lines. With a discrete evolution family \(\Phi = \{\Phi(m,n)\}_{(m,n) \in \Delta}\) associate the subspace \(X_1 = \{x \in X : \Phi(\cdot,0)x \in \ell^\infty(\mathbb{N},X)\}\). Supposing that \(X_1\) is closed and complemented, he proves that the admissibility of the pair \((\ell^\infty(\mathbb{N},X), \ell_0^1(\mathbb{N},X))\) implies the uniform dichotomy of \(\Phi\). Under the same hypothesis on \(X_1\), he obtains that the admissibility of the pair \((\ell^\infty(\mathbb{N},X), \ell_0^p(\mathbb{N},X))\) with \(p \in (1,\infty]\) is a sufficient condition for an exponential dichotomy of \(\Phi\), which becomes necessary when \(\Phi\) has exponential growth. The author gives characterizations for exponential dichotomy of evolution families in terms of the solvability of associated difference and integral equations.
reviewed by: S. Siegmund
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 5077922
zbMATH Keywords: uniform dichotomy
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: discrete evolution family
zbMATH Keywords: evolution family
zbMATH Keywords: difference equation
zbMATH Keywords: integral equation
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.12.002
OpenAlex ID: W2014202711
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DOI: 10.1016/J.JMAA.2005.12.002
description: scientific article; zbMATH DE number 5077922