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

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
Received 3 February 2007; Revised 18 September 2007; Accepted 27 November 2007

Recommended by H. Bevan Thompson

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.