The behavior of positive solutions of a nonlinear second-order difference equation
type: Item
instance of: scholarly article
title: The behavior of positive solutions of a nonlinear second-order difference equation
zbMATH Open document ID: 1146.39018
DOI: 10.1155/2008/653243
author: Stevo Stević
author: Kenneth S. Berenhaut
published in: Abstract and Applied Analysis
publication date: 2008-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/54553
review text: Summary: The authors study the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation \(x_{n}=f(x_{n - 2})/g(x_{n - 1})4, 4n\in \mathbb N_{0}\), where \(f,g\in C[(0,\infty ),(0,\infty )]\). It is shown that if \(f\) and \(g\) are nondecreasing, then for every solution of the equation the subsequences \(\{x_{2n}\}\) and \(\{x_{2n - 1}\}\) are eventually monotone. For the case when \(f(x)=\alpha +\beta x\) and \(g\) satisfies the conditions \(g(0)=1\), \(g\) is nondecreasing, and \(x/g(x)\) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then \(f(x) = c_{1}/x\) and \(g(x) = c_{2}x\), for some positive \(c_{1}\) and \(c_{2}\).
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 5313161
Wikidata QID: Q58644381
MaRDI profile type: Publication
OpenAlex ID: W2034590304
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description: scientific article; zbMATH DE number 5313161