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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 391265, 15 pages
http://dx.doi.org/10.1155/2008/391265
Research Article

On the Rational Recursive Sequence π‘₯ 𝑛 + 1 = ( 𝛼 βˆ’ 𝛽 π‘₯ 𝑛 ) / ( 𝛾 βˆ’ 𝛿 π‘₯ 𝑛 βˆ’ π‘₯ 𝑛 βˆ’ π‘˜ )

1Mathematics Department, Faculty of Science, Taif University, El-Taif 5700, El-Hawiyah, Kingdom of Saudi Arabia
2Mathematics Department, Faculty of Science, Zagazig University, Zagazig 4419, Egypt
3Mathematics Department, Faculty of Science, El-Minia University, El Minia 61519, Egypt

Received 1 November 2007; Accepted 11 May 2008

Academic Editor: Attila Gilanyi

Copyright © 2008 E. M. E. Zayed 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 study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation π‘₯ 𝑛 + 1 = ( 𝛼 βˆ’ 𝛽 π‘₯ 𝑛 ) / ( 𝛾 βˆ’ 𝛿 π‘₯ 𝑛 βˆ’ π‘₯ 𝑛 βˆ’ π‘˜ ) , 𝑛 = 0 , 1 , 2 , … , π‘˜ ∈ { 1 , 2 , … } , in the two cases: (i) 𝛿 β‰₯ 0 , 𝛼 > 0 , 𝛾 > 𝛽 > 0 ; (ii) 𝛿 β‰₯ 0 , 𝛼 = 0 , 𝛾 , 𝛽 > 0 , where the coefficients 𝛼 , 𝛽 , 𝛾 , and 𝛿 , and the initial conditions π‘₯ βˆ’ π‘˜ , π‘₯ βˆ’ π‘˜ + 1 , … , π‘₯ βˆ’ 1 , π‘₯ 0 are real numbers. We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients of this equation.