Global behavior of the difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\)
type: Item
instance of: scholarly article
title: Global behavior of the difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\)
zbMATH Open document ID: 1474.39026
DOI: 10.1155/2014/705893
author: Hongjian Xi
author: Taixiang Sun
author: Bin Qin
author: Hui Wu
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: We consider the following difference equation \(x_{n + 1} = x_{n - 1} g(x_n)\), \(n = 0,1, \ldots\), where initial values \(x_{- 1}, x_0 \in [0, + \infty)\) and \(g : [0, + \infty) \rightarrow(0,1]\) is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) for some \(a \in [0, + \infty)\). (2) Assume \(a \in(0, + \infty)\). Then the set of initial conditions \((x_{- 1}, x_0) \in(0, + \infty) \times(0, + \infty)\) such that the positive solutions of this equation converge to \(a, 0, a, 0, \ldots\), or \(0, a, 0, a, \ldots\) is a unique strictly increasing continuous function or an empty set.
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A20
zbMATH DE Number: 7022916
MaRDI profile type: Publication
Wikidata QID: Q59040773
full work available at URL: https://doi.org/10.1155/2014/705893
OpenAlex ID: W1936022419
cites work: Convergence of solutions in a nonhyperbolic case.
cites work: Dynamics of a family of nonlinear delay difference equations
cites work: Global attractivity of a family of nonautonomous max-type difference equations
cites work: On the asymptotic behavior of a difference equation with maximum
cites work: Global stability of a max-type difference equation
cites work: Dynamics of the max-type difference equation \({x_n=\max\{\frac{1}{x_{n-m}},\frac{A_n}{x_{n-r}}\}}\)
cites work: On boundedness of solutions of the difference equation \(x_{n+1}=p+\frac{x_{n-1}}{x_{n}}\) for \(p<1\)
cites work: Dynamics of second order rational difference equations. With open problems and conjectures
cites work: Convergence properties in the nonhyperbolic case \(x_{n+1} = \frac {x_{n-1}}{1+f(x_{n})}\)
description: scientific article; zbMATH DE number 7022916