Abstract and Applied Analysis
Volume 2011 (2011), Article ID 561682, 8 pages
http://dx.doi.org/10.1155/2011/561682
Research Article

Unbounded Solutions of the Difference Equation 𝑥 𝑛 = 𝑥 𝑛 − 𝑙 𝑥 𝑛 − 𝑘 − 1

Stevo Stević1 and Bratislav Iričanin2

1Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia
2Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11120, Serbia

Received 14 April 2011; Revised 22 June 2011; Accepted 22 June 2011

Academic Editor: Josef Diblík

Copyright © 2011 Stevo Stević and Bratislav 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

The following difference equation 𝑥 𝑛 = 𝑥 𝑛 − 𝑙 𝑥 𝑛 − 𝑘 − 1 , 𝑛 ∈ ℕ 0 , where 𝑘 , 𝑙 ∈ ℕ , 𝑘 < 𝑙 , g c d ( 𝑘 , 𝑙 ) = 1 , and the initial values 𝑥 − 𝑙 , … , 𝑥 − 2 , 𝑥 − 1 are real numbers, has been investigated so far only for some particular values of 𝑘 and 𝑙 . To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameters 𝑘 and 𝑙 , where the initial values satisfy the following condition 𝑥 m i n − 𝑙 , … , 𝑥 − 2 , 𝑥 − 1 .