Integral conditions for exponential dichotomy: a nonlinear approach
type: Item
instance of: scholarly article
title: Integral conditions for exponential dichotomy: a nonlinear approach
zbMATH Open document ID: 1196.34066
author: Bogdan Sasu
published in: Bulletin des Sciences Mathématiques
publication date: 2010-04-27T00:00:00Z
review text: This paper gives necessary and sufficient conditions for linear skew-product semiflows to be exponentially dichotomic. The mentioned skew-product semiflows \((\Phi,\sigma)\) are defined on the product \(\Theta\times X\) of a metric space \((\Theta,d)\) and a Banach space \(X\); applications to infinite-dimensional evolutionary equations are indicated. In this framework a characterization of exponential dichotomies is derived in terms of related results on uniform exponential stability due to \textit{S. Rolewicz} [J. Math. Anal. Appl. 115, 434--441 (1986; Zbl 0597.34064)]. The latter involve certain integrability conditions on \(N(\|\Phi(\theta,\tau)x\|)\), where \(\Phi\) is the evolution operator and \(N\) denotes a continuous nondecreasing positive definite function. An application to evolution families is given and corresponding results for discrete skew-product flows can be found in [\textit{B. Sasu}, Discrete Dyn. Nat. Soc. 2009, Article ID 324273, 18 p. (2009; Zbl 1178.39029)].
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 5702055
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: projection family
zbMATH Keywords: skew-product flow
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.bulsci.2009.06.006
OpenAlex ID: W1966223985
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DOI: 10.1016/J.BULSCI.2009.06.006
description: scientific article; zbMATH DE number 5702055