Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 39404, 7 pages
doi:10.1155/2007/39404
Research Article

On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj

Stevo Stević

Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 35/I, Belgrade 11001, Serbia

Received 28 September 2006; Revised 3 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 give a complete picture regarding the behavior of positive solutions of the following important difference equation: xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj, n∈ℕ0, where αi, i∈{1,…,k}, and βj, j∈{1,…,m}, are positive numbers such that ∑i=1kαi=∑j=1mβj=1, and pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural numbers such that p1<p2<⋯<pk and q1<q2<⋯<qm. The case when gcd(p1,…,pk,q1,…,qm)=1 is the most important. For the case we prove that if all pi, i∈{1,…,k}, are even and all qj, j∈{1,…,m}, are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.