Discrete Dynamics in Nature and Society
Volume 2007 (2007), Article ID 40963, 9 pages
doi:10.1155/2007/40963
Research Article
On the Recursive Sequence xn+1=A+xnp/xn−1r
Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000, Serbia
Received 1 October 2007; Accepted 5 November 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
The paper considers the boundedness character of positive solutions of the difference equation xn+1=A+xnp/xn−1r, n∈ℕ0, where A, p, and r are positive real numbers. It is shown that (a) If p2≥4r>4, or p≥1+r, r≤1, then this equation has positive unbounded solutions; (b) if p2<4r, or 2r≤p<1+r, r∈(0,1), then all positive solutions of the equation are bounded. Also, an analogous result is proved
regarding positive solutions of the max type difference equation xn+1=max{A,xnp/xn−1r}, where A, p, q∈(0,∞).