On the behaviour of solutions of the difference equation \(x_{n+1}=\alpha+ \frac{x_{n-2}}{x_{n-1}} + \frac{x_{n-1}}{x_n}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.39010
author: Vu Van Khuong
author: Tran Hong Thai
publication date: 2012-08-28T00:00:00Z
full work available at URL: index.html
Mathematics Subject Classification ID: 39A20
Mathematics Subject Classification ID: 39A21
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A30
zbMATH DE Number: 6073161
zbMATH Keywords: boundedness
zbMATH Keywords: global attractivity
zbMATH Keywords: nonoscillatory solution
zbMATH Keywords: rational difference equation
zbMATH Keywords: stability
zbMATH Keywords: monotonicity
MaRDI profile type: Publication
title: On the behaviour of solutions of the difference equation \(x_{n+1}=\alpha+ \frac{x_{n-2}}{x_{n-1}} + \frac{x_{n-1}}{x_n}\)
published in: International Mathematical Forum
review text: The authors consider the third order nonlinear difference equation NEWLINE\[NEWLINE x_{n+1} = \alpha + {{x_{n-2}}\over{x_{n-1}}} + {{x_{n-1}}\over{x_n}} NEWLINE\]NEWLINE with \(\alpha\geq 0\), \(x_{-2},\;x_{-1},\;x_0\in(0,\infty)\). By setting \(y_{n-1}= x_{n-1}/x_n\) the following difference equation is associated NEWLINE\[NEWLINE \displaystyle{y_{n+1} = {{\alpha+y_{n-1}+y_{n-1}}\over{\alpha+y_n+y_{n-1}}}} NEWLINE\]NEWLINE with the unique equilibrium \(\bar{y}=1\). By setting NEWLINE\[NEWLINE z_{n+1} = {{1}\over{\alpha}}(\alpha/2+y_{n+1}) NEWLINE\]NEWLINE there is associated the normalized equation NEWLINE\[NEWLINE \displaystyle{z_{n+1} = {{z_n+\gamma z_{n-1} + \delta z_{n-2}}\over {2(z_n+cz_{n-1})}}}. NEWLINE\]NEWLINE The paper considers various qualitative behavior of the solutions of these equations (stability, monotonicity, boundedness).
reviewed by: Vladimir Răsvan
description: scientific article; zbMATH DE number 6073161