Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 91292, 7 pages
doi:10.1155/2007/91292
Research Article

A Global Convergence Result for a Higher Order Difference Equation

Bratislav D. Iričanin

Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Beograd 11000, Serbia

Received 9 February 2007; Revised 15 April 2007; Accepted 13 June 2007

Copyright © 2007 Bratislav D. Iričanin. 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

Let f(z1,…,zk)∈C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of ℝ, and k∈ℕ. Assume that f(y1,y2,…,yk)≥f(y2,…,yk,y1) if y1≥max{y2, …,yk}, f(y1,y2,…,yk)≤f(y2,…,yk,y1) if y1≤min{y2,…,yk}, and f is non- decreasing in the last variable zk. We then prove that every bounded solution of an autonomous difference equation of order k, namely, xn=f(xn−1,…,xn−k), n=0,1,2,…, with initial values x−k,…,x−1∈I, is convergent, and every unbounded solution tends either to +∞ or to −∞.