`Discrete Dynamics in Nature and SocietyVolume 2007, Article ID 13737, 9 pageshttp://dx.doi.org/10.1155/2007/13737`
Research Article

## Asymptotic Periodicity of a Higher-Order Difference Equation

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

Received 27 April 2007; Accepted 13 September 2007

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 give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn=f(xnp1,,xnpk,xnq1,,xnqm), n0, where pi, i{1,,k}, and qj, j{1,,m}, are natural numbers such that p1<p2<<pk, q1<q2<<qm and gcd(p1,,pk,q1,,qm)=1, the function fC[(0,)k+m, (α,)], α>0, is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function gC[(α,),(α,)] such that g(g(x))=x, x(α,), x=f(x,,xk,g(x),,g(x)m), x(α,), limxα+g(x)=+, and limx+g(x)=α. It is proved that if all pi, i{1,,k}, are even and all qj, j{1,,m} are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.