Advances in Difference Equations
Volume 2008 (2008), Article ID 143723, 6 pages
doi:10.1155/2008/143723

The Periodic Character of the Difference Equation xn+1=f(xn−l+1,xn−2k+1)

Taixiang Sun1 and Hongjian Xi2

1Department of Mathematics, College of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, China
 2Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003, Guangxi, China

Abstract

In this paper, we consider the nonlinear difference equation xn+1=f(xn−l+1,xn−2k+1), n=0,1,…, where k,l∈{1,2,…} with 2k≠l and gcd(2k,l)=1 and the initial values x−α,x−α+1,…,x0∈(0,+∞) with α=max{l−1,2k−1}. We give sufficient conditions under which every positive solution of this equation converges to a ( not necessarily prime ) 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.