A criterion for the exponential stability of linear difference equations.
type: Item
instance of: scholarly article
title: A criterion for the exponential stability of linear difference equations.
zbMATH Open document ID: 1068.39019
author: Mihály Pituk
published in: Applied Mathematics Letters
publication date: 2005-03-07T00:00:00Z
review text: Based on a result by \textit{B. Aulbach} and \textit{Nguyen Van Minh} [J. Difference Eq. Appl.~2, 251--262 (1996; Zbl 0880.39009)] the author proves the following stability criterion for the linear nonautonomous difference equation \[ x_{n+1}=A_nx_n, \] where each \(A_n\), \(n\in{\mathbb N}\), is a bounded linear operator on a real or complex Banach space, and one has \(\sup_{n\in{\mathbb N}}| A_n| <\infty\): The above linear equation is exponentially stable if and only if for every \(p\)-summable sequence \(f_n\), \(1<p<\infty\), the solution of the inhomogeneous initial value problem \(x_{n+1}=A_nx_n+f_n\), \(x_1=0\) is bounded.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 2140742
zbMATH Keywords: difference equations in Banach space
zbMATH Keywords: linear equations
zbMATH Keywords: \(l_p\)-spaces
zbMATH Keywords: exponential
zbMATH Keywords: stability
zbMATH Keywords: bounded linear operator
zbMATH Keywords: initial value problem
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: j.aml.2004.06.005
OpenAlex ID: W2091098051
cites work: The concept of spectral dichotomy for linear difference equations II
cites work: Q5791753
cites work: Q4774403
cites work: Q5533446
DOI: 10.1016/J.AML.2004.06.005
Recommended article: Criteria for the exponential stability of linear evolution difference equations
Recommended article: Q3512037
Recommended article: Exponential Stability in Non-linear Difference Equations
Recommended article: On exponential stability of linear difference-differential systems
Recommended article: Exponential stability of some linear continuous time difference systems
Recommended article: Q5290082
Recommended article: Exponential stability of linear delay difference equations with continuous time
Recommended article: Q3693965
Recommended article: New characterizations of exponential dichotomy and exponential stability of linear difference equations
Recommended article: Sufficient conditions for the exponential stability of nonautonomous difference equations
description: scientific article; zbMATH DE number 2140742