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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 534947, 15 pages
Research Article

Dynamics of a Higher-Order Nonlinear Difference Equation

1School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, China
2Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, China
3Department of Mathematics, Hexi University, Zhangye, Gansu 734000, China

Received 20 April 2010; Accepted 18 July 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Guo-Mei Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the higher-order nonlinear difference equation 𝑥 𝑛 + 1 = ( 𝛼 + 𝑥 𝑛 ) / ( 𝐴 + 𝐵 𝑥 𝑛 + 𝑥 𝑛 𝑘 ) , 𝑛 = 0 , 1 , , where parameters are positive real numbers and initial conditions 𝑥 𝑘 , , 𝑥 0 are nonnegative real numbers, 𝑘 2 . We investigate the periodic character, the invariant intervals, and the global asymptotic stability of all positive solutions of the abovementioned equation. We show that the unique equilibrium of the equation is globally asymptotically stable under certain conditions.