Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 258502, 12 pages
http://dx.doi.org/10.1155/2012/258502
Research Article

Some Notes on the Difference Equation 𝑥 𝑛 + 1 = 𝛼 + ( 𝑥 𝑛 − 1 / 𝑥 𝑘 𝑛 )

Mehmet Gümüş1 and Özkan Öcalan2

1Department of Mathematics, Faculty of Arts and Sciences, Bülent Ecevit University, 67100 Zonguldak, Turkey
2Department of Mathematics, Faculty of Science and Literature, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 12 April 2012; Accepted 20 May 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 Mehmet Gümüş and Özkan Öcalan. 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 investigate the behavior of the solutions of the recursive sequence 𝑥 𝑛 + 1 = 𝛼 + 𝑥 𝑛 − 1 / 𝑥 𝑘 𝑛 , 𝑛 = 0 , 1 , … , where 𝛼 ∈ [ 0 , ∞ ) , 𝑘 ∈ ( 0 , ∞ ) , and the initial conditions 𝑥 − 1 , 𝑥 0 are arbitrary positive numbers. Included are results that considerably improve those in the recently published paper by Hamza and Morsy (2009).