Advances in Difference Equations
Volume 2006 (2006), Article ID 27637, 7 pages
doi:10.1155/ADE/2006/27637
Global behavior of a higher-order rational difference equation
Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530004, China
Received 17 January 2006; Revised 6 April 2006; Accepted 12 April 2006
Copyright © 2006 Hongjian Xi and Taixiang Sun. 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.
Abstract
We investigate in this paper the global behavior of the following
difference equation: xn+1=(Pk(xn−i0,xn−i1,…,xn−i2k)+b)/(Qk(xn−i0,xn−i1,…,xn−i2k)+b), n=0,1,…, under appropriate assumptions,
where b∈[0,∞), k≥1, i0,i1,…,i2k∈{0,1,…} with i0<i1<…<i2k, the initial conditions xi−2k,xi−2k+1,…,x0∈(0,∞). We prove that
unique equilibrium x¯=1 of that equation is globally
asymptotically stable.