## Abstract and Applied Analysis

### Global Behavior of the Difference Equation ${x}_{n+1}={x}_{n-1}g({x}_{n})$

#### Abstract

We consider the following difference equation ${x}_{n+1}={x}_{n-1}g({x}_{n})$, $n=0,1,\dots ,$ where initial values ${x}_{-1},{x}_{0}\in [0,+\infty )$ and $g:[0,+\infty )\to (0,1]$ is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges to $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ 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 $a,0,a,0,\dots ,$ or $0,a,0,a,\dots$ is a unique strictly increasing continuous function or an empty set.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 705893, 5 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049104

Digital Object Identifier
doi:10.1155/2014/705893

Mathematical Reviews number (MathSciNet)
MR3166646

Zentralblatt MATH identifier
07022916

#### Citation

Xi, Hongjian; Sun, Taixiang; Qin, Bin; Wu, Hui. Global Behavior of the Difference Equation ${x}_{n+1}={x}_{n-1}g({x}_{n})$. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 705893, 5 pages. doi:10.1155/2014/705893. https://projecteuclid.org/euclid.aaa/1425049104

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