Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation: x n + 1 = f ( x n − r 1 , … , x n − r k ) g ( x n − m 1 , … , x n − m l ) + 1 f ( x n − r 1 , … , x n − r k ) + g ( x n − m 1 , … , x n − m l ) , n = 0 , 1 , … , where f ∈ C ( ( 0 , + ∞ ) k , ( 0 , + ∞ ) ) and g ∈ C ( ( 0 , + ∞ )...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical analysis and applications 2007-06, Vol.330 (1), p.462-466
Main Authors: Sun, Taixiang, Xi, Hongjian
Format: Article
Language:English
Subjects:
Difference equation
Equilibrium
Exact sciences and technology
Finite differences and functional equations
Global asymptotic stability
Mathematical analysis
Mathematics
Sciences and techniques of general use
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this note, we consider the following rational difference equation: x n + 1 = f ( x n − r 1 , … , x n − r k ) g ( x n − m 1 , … , x n − m l ) + 1 f ( x n − r 1 , … , x n − r k ) + g ( x n − m 1 , … , x n − m l ) , n = 0 , 1 , … , where f ∈ C ( ( 0 , + ∞ ) k , ( 0 , + ∞ ) ) and g ∈ C ( ( 0 , + ∞ ) l , ( 0 , + ∞ ) ) with k , l ∈ { 1 , 2 , … } , 0 ⩽ r 1 < ⋯ < r k and 0 ⩽ m 1 < ⋯ < m l , and the initial values are positive real numbers. We give sufficient conditions under which the unique equilibrium x ¯ = 1 of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the recent literature.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2006.07.096