On the period five trichotomy of all positive solutions of \(x_{n+1}=(p+x_{n-2})/x_{n}\).
type: Item
instance of: scholarly article
title: On the period five trichotomy of all positive solutions of \(x_{n+1}=(p+x_{n-2})/x_{n}\).
zbMATH Open document ID: 1040.39003
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: The authors study the asymptotic behavior of positive solutions of the difference equation in the title, where \(p\) is a positive real parameter. For any \(p>0\) there exists a unique positive equilibrium \(\bar{x}=(1+\sqrt{1+4p})/2\). When \(p=1\) there exists a cycle of period five. By exploiting some special properties of the equation the authors show that if \(0<p<1\) or \(p\geq2\) every positive bounded solution converges to \(\bar{x}\). When \(0 < p < 1\) the existence of unbounded solutions is proved. When \(p\geq2\) the positive equilibrium is globally asymptotically stable. The authors make a conjecture: if \(1<p<2\), then the positive equilibrium \(\bar{x}\) is globally asymptotically stable.
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 39A20
zbMATH DE Number: 2056494
zbMATH Keywords: global stability
zbMATH Keywords: periodic solution
zbMATH Keywords: multiplicative difference equation
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: bounded solution
zbMATH Keywords: unbounded solutions
zbMATH Keywords: positive equilibrium
author: Elias Camouzis
author: Richard DeVault
author: Witold A. Kosmala
reviewed by: Oleg Anashkin
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.016
OpenAlex ID: W2069616740
cites work: Three Trichotomy Conjectures
cites work: On Period-Two Convergence in Rational Equations
cites work: A rational difference equation
cites work: On the Asymptotic Stability of a Class of Linear Difference Equations
DOI: 10.1016/J.JMAA.2003.10.016
description: scientific article; zbMATH DE number 2056494