IJPAM: Volume 52, No. 5 (2009)

THE DYNAMICAL BEHAVIOR OF A CLASS OF HIGH
ORDER NONLINEAR RATIONAL DIFFERENCE EQUATIONS

Zhi-Long Jin
Department of Mathematics
School of Information Engineering
Lanzhou Commercial College
Lanzhou, Gansu, 730020, P.R. CHINA
e-mail: [email protected]


Abstract.The purpose of this note is to investigate the global stability of the positive equilibrium of the following high order nonlinear rational difference equation:

\begin{displaymath}
x_{n+1}=\frac{x_{n-i}x_{n-j}x_{n-k}+x_{n-i}+x_{n-j}+x_{n-k}+a}{%
x_{n-i}x_{n-j}+x_{n-i}x_{n-k}+x_{n-j}x_{n-k}+1+a}\,,
\end{displaymath}

where $n=0, 1, ... , i$, $j$, $k\in N$ are fixed. A simple and short proof for the globally asymptotically stability of positive equilibrium is shown and our result including the results of the fourth-order rational difference equations in [#!3!#,#!4!#,#!5!#,#!zhang!#]. Further more, the same result is also obtained for a more general difference equations.

Received: October 10, 2007

AMS Subject Classification: 39A10

Key Words and Phrases: high order nonlinear difference equation, positive equilibrium, global asymptotic stability

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 5