Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 23503, 10 pages
doi:10.1155/2007/23503
Research Article

On Global Periodicity of a Class of Difference Equations

Stevo Stević

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia

Received 16 October 2006; Revised 6 December 2006; Accepted 11 December 2006

Copyright © 2007 Stevo Stević. 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 show that the difference equation xn=f3(xn−1)f2(xn−2)f1(xn−3), n∈ℕ0, where fi∈C[(0,∞),(0,∞)], i∈{1,2,3}, is periodic with period 4 if and only if fi(x)=ci/x for some positive constants ci, i∈{1,2,3} or if fi(x)=ci/x when i=2 and fi(x)=cix if i∈{1,3}, with c1c2c3=1. Also, we prove that the difference equation xn=f4(xn−1)f3(xn−2)f2(xn−3)f1(xn−4), n∈ℕ0, where fi∈C[(0,∞),(0,∞)], i∈{1,2,3,4}, is periodic with period 5 if and only if fi(x)=ci/x, for some positive constants ci, i∈{1,2,3,4}.