Asymptotic behavior of a forced difference equation
type: Item
instance of: scholarly article
title: Asymptotic behavior of a forced difference equation
zbMATH Open document ID: 0860.39011
author: John R. Graef
author: Chuanxi Qian
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-12-09T00:00:00Z
review text: The aim of this paper is to study the asymptotic behavior of the forced difference equation (1) \(x_{n+1} -x_n+ px_{n-k} =r_n\), \(n=0,1, \dots\), where \(p\) is a real number, \(k\) is a positive integer, and \(\{r_n\}\) is a real sequence. The corresponding homogeneous equation of (1) is (2) \(y_{n+1} -y_n+ py_{n-k} =0\), \(n=0,1, \dots\). It is well known that if \(\lambda_0\) is a simple real root of the characteristic equation for (2) with the largest modulus among all the roots, then the asymptotic behavior of solutions of (2) is dominated by the solution \(\{\lambda^n\}\). In the present paper it is shown that under certain hypotheses, the solutions of (1) have the same property. Moreover, for any solution \(\{x_n\}\) of (1), \(\lim_{n\to \infty} \{\lambda_0^{-n} x_n\}\) is found explicitly in terms of the initial conditions of \(\{x_n\}\). In the case \(p>0\) a necessary and sufficient condition is established for every solution of (1) to tend to zero.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 944361
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: forced difference equation
reviewed by: Alexander Mednykh
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0387
OpenAlex ID: W2045740756
DOI: 10.1006/JMAA.1996.0387
description: scientific article; zbMATH DE number 944361