Polynomial contractions and Lyapunov regularity
type: Item
instance of: scholarly article
title: Polynomial contractions and Lyapunov regularity
zbMATH Open document ID: 1196.37050
author: Luís Barreira
author: Claudia Valls
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-06-09T00:00:00Z
review text: The article deals with a nonautonomous linear dynamics with discrete time for systems defined by the product of a sequence of matrices \((A_n)_{n \in {\mathbb N}}\) (\(x_{n+1} = A_nx_n\), \(n = 0,1,\dots\)) and satisfying the estimate \[ \|A_{m-1} \cdots A_n\| \leq D(m / n)^\lambda n^a, \quad m \geq n \] for some \(\lambda < 0\), \(D \geq 0\), \(a \geq 0\). If \(a = 0\) then such a system admit a \textit{uniform polynomial contraction}; if \(a > 0\) then such a system admits a \textit{nonuniform polynomial contraction}. In the article, it is presented necessary and sufficient conditions under that the system \((A_n)_{n \in {\mathbb N}}\) admits nonuniform polynomial contraction in terms of ``polynomial'' Lyapunov exponents, sharp lower and upper bounds for constants \(\lambda\) and \(a\) in terms of matrices \(A_n\), and some results about the stability of systems with nonuniform polynomial behavior under small linear and nonlinear perturbations. Unfortunately, in the article there is no examples of systems admitting uniform and nonuniform polynomial contraction.
Mathematics Subject Classification ID: 37C60
Mathematics Subject Classification ID: 34D08
zbMATH DE Number: 5718402
zbMATH Keywords: difference equations
zbMATH Keywords: Lyapunov regularity
zbMATH Keywords: uniform polynomial behavior
zbMATH Keywords: nonuniform polynomial behavior
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.na.2010.03.013
OpenAlex ID: W1971118889
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DOI: 10.1016/J.NA.2010.03.013
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description: scientific article; zbMATH DE number 5718402