Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 475038, 8 pages
http://dx.doi.org/10.1155/2012/475038
Research Article

The Dynamics of the Solutions of Some Difference Equations

H. El-Metwally,1,2 R. Alsaedi,1 and E. M. Elsayed2,3

1Department of Mathematics, College of Science and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia
2Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 22 December 2011; Revised 17 May 2012; Accepted 17 May 2012

Academic Editor: Ibrahim Yalcinkaya

Copyright © 2012 H. El-Metwally 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

This paper is devoted to investigate the global behavior of the following rational difference equation: 𝑦 𝑛 + 1 = 𝛼 𝑦 𝑛 − 𝑡 ∑ / ( 𝛽 + 𝛾 𝑘 𝑖 = 0 𝑦 𝑝 𝑛 − ( 2 𝑖 + 1 ) ∏ 𝑘 𝑖 = 0 𝑦 𝑞 𝑛 − ( 2 𝑖 + 1 ) ) , 𝑛 = 0 , 1 , 2 , … , where 𝛼 , 𝛽 , 𝛾 , 𝑝 , 𝑞 ∈ ( 0 , ∞ ) and 𝑘 , 𝑡 ∈ { 0 , 1 , 2 , … } with the initial conditions 𝑥 0 , 𝑥 − 1 , … , 𝑥 − 2 𝑘 , 𝑥 − 2 m a x { 𝑘 , 𝑡 } − 1 ∈ ( 0 , ∞ ) . We will find and classify the equilibrium points of the equations under studying and then investigate their local and global stability. Also, we will study the oscillation and the permanence of the considered equations.