Two classes of piecewise-linear difference equations with eventual periodicity three
type: Item
instance of: scholarly article
title: Two classes of piecewise-linear difference equations with eventual periodicity three
zbMATH Open document ID: 1135.39003
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-23T00:00:00Z
review text: The author proves two of the conjectures stated in his previous paper [ibid. 295, No. 2, 570--575 (2004; Zbl 1058.39002)]. More precisely, he proves the following two theorems. Let \(\alpha\) be an integer greater than 2 and let \(\{x_n\}\) be a nontrivial solution of the equation \(x_{n+1}=(Ax)_n\), where \((Ax)_n=(x_n+x_{n-1})/\alpha\) if \(\alpha | x_n+x_{n-1}\) and \((Ax)_n=-x_n-x_{n-1}\) otherwise. Then \(\{x_n\}\) is eventually periodic of prime period 3. Let \(\{x_n\}\) be a nontrivial solution of the equation \(x_{n+1}=(Bx)_n\), where \((Bx)_n=(x_n-x_{n-1})/3\) if \(3 | x_n-x_{n-1}\) and \((Bx)_n=-x_n-x_{n-1}\) otherwise. Then \(\{x_n\}\) is eventually periodic of prime period 3.
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 5156447
zbMATH Keywords: piecewise-linear difference equation
zbMATH Keywords: periodic solution
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: recursive sequences
author: Jeffrey Feuer
reviewed by: Pavel Rehak
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.018
OpenAlex ID: W2075349267
cites work: On some difference equations with eventually periodic solutions
cites work: Some Equations with Eventually Periodic Solutions†
cites work: Two classes of piecewise-linear difference equations with eventual periodicity six
DOI: 10.1016/J.JMAA.2006.10.018
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description: scientific article; zbMATH DE number 5156447