Stability of a linear difference equation and estimation of its fundamental solution
type: Item
instance of: scholarly article
title: Stability of a linear difference equation and estimation of its fundamental solution
zbMATH Open document ID: 1268.39016
DOI: 10.3103/S1066369X11120048
author: A. Yu. Kulikov
author: V. V. Malygina
published in: Russian Mathematics
publication date: 2012-11-20T00:00:00Z
review text: The authors analyze different types of stability for the following non-autonomous difference equation with delays \[ x(n+1)-x(n) + \sum_{k=0}^{N} A_{k}(n)x(n-\alpha_k(n))=f(n),\tag{1} \] where \(m\in\{0,1,\dots\}\), \(n\in\mathbb{N}_{m}:=\{m,m+1,\dots\}\), \(f:\mathbb{N}_0\rightarrow \mathbb{C}^{r}\), \(A_k: \mathbb{N}_0\rightarrow \mathbb{C}^{r\times r}\) are matrix functions and \(\alpha_k:\mathbb{N}_0\rightarrow \mathbb{N}_0\), \(k=0,\dots,N\). The authors impose the so-called \(V\)-condition, namely, the quantity \(V:=\sup_{n\in\mathbb{N}_0}\sum_{j=n-\alpha(n)}^{n} a(i)\) is finite (here \(\alpha\) denotes \(\max_{k}\{\alpha_k\}\) and \(a(n)=\sum_{k=0}^{N}|A_k(n)|\) if \(n\geq 0\) and \(a(n)=0\) otherwise). The main results of the paper are the following: 1. If \(f\equiv 0\) and \(m\geq 0\), assuming the \(V\)-condition they characterize different notions of stability (Lyapunov, uniform, asymptotical, uniformly asymptotical and exponential) in terms of the associate fundamental solution \(K(n,m)\), \(n\geq m\). 2. If \(m=0\) in \((1)\), and the zero initial function is considered, the stability properties of (1) are studied with respect to certain linear normed spaces of functions. To this end the authors consider an operator of (1) which is strongly linked with the Cauchy function appearing in a related functional differential equation. 3. Finally, they apply the results of the second point to establish a sufficient condition which ensures the exponential stability of \(x(n+1)-x(n) + A x(n-\alpha(n))=f(n)\), \(n\in\mathbb{N}_0\), where \(A\) is a constant \(\mathbb{C}^{r\times r}\) matrix.
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A06
zbMATH DE Number: 6108756
zbMATH Keywords: fundamental solution
zbMATH Keywords: effective conditions
zbMATH Keywords: Cauchy function
zbMATH Keywords: Bohl-Perron theorems
zbMATH Keywords: linear normed spaces of functions
zbMATH Keywords: Cauchy operator
zbMATH Keywords: eigenvalue
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: uniform stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: non-autonomous difference equation with delays
zbMATH Keywords: exponential stability
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6108756