Abstract and Applied Analysis

Global Behavior of the Difference Equation x n + 1 = x n - 1 g ( x n )

Hongjian Xi, Taixiang Sun, Bin Qin, and Hui Wu

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Abstract

We consider the following difference equation x n + 1 = x n - 1 g ( x n ) , n = 0,1 , , where initial values x - 1 , x 0 [ 0 , + ) and g : [ 0 , + ) ( 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 , , or 0 , a , 0 , a , for some a [ 0 , + ) . (2) Assume a ( 0 , + ) . Then the set of initial conditions ( x - 1 , x 0 ) ( 0 , + ) × ( 0 , + ) such that the positive solutions of this equation converge to a , 0 , a , 0 , , or 0 , a , 0 , a , 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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