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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 583230, 13 pages
http://dx.doi.org/10.1155/2010/583230
Research Article

On the Recursive Sequence 𝑥 𝑛 + 𝟏 = m a x { 𝑥 𝑛 , 𝐴 } / 𝑥 𝟐 𝑛 𝑥 𝑛 𝟏

Mathematics Department, Ahmet Kelesoglu Education Faculty, Selcuk University, Meram Yeni Yol, 42090 Konya, Turkey

Received 17 March 2010; Accepted 11 May 2010

Academic Editor: Guang Y. Zhang

Copyright © 2010 Ibrahim Yalcinkaya et al. 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 periodic nature of the solution of the max-type difference equation 𝑥 𝑛 + 1 = m a x { 𝑥 𝑛 , 𝐴 } / 𝑥 2 𝑛 𝑥 𝑛 1 , 𝑛 = 0 , 1 , 2 , , where the initial conditions are 𝑥 1 = 𝐴 𝑟 1 and 𝑥 0 = 𝐴 𝑟 2 for 𝐴 ( 0 , ) , and that 𝑟 1 and 𝑟 2 are positive rational numbers. The results in this paper solve the Open Problem proposed by Grove and Ladas (2005).