Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 475038, 8 pages doi: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. Elsayed 2, 3 1 Department of Mathematics, College of Science and Arts, King Abdulaziz University, Rabigh Campus, P.O. Box 344, Rabigh 21911, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt 3 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to E. M. Elsayed, emmelsayed@yahoo.com Received 22 December 2011; Revised 17 May 2012; Accepted 17 May 2012 Academic Editor: Ibrahim Yalcinkaya Copyright q 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. This paper is devoted to investigate the global behavior of the following rational difference equation: y n1  αy n-t /β  γ ∑ k i0 y p n-2i1  k i0 y q n-2i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 -2k ,x -2 max{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. 1. Introduction The aim of this paper is to study the dynamics of the solutions of the following recursive sequence: y n1  αy n-t β  γ ∑ k i0 y p n-2i1  k i0 y q n-2i1 , n  0, 1, 2,..., 1.1 where α,β,γ,p,q ∈ 0, ∞ and K ∈ {0, 1, 2,...}, where K  max{k,t}, with the initial conditions x 0 , x -1 ,..., x -2k ,x -2K-1 ∈ 0,∞. We deal with the classification of the equilibrium points of 1.1 in terms of being stable or unstable, where we investigate the global attractor of the solutions of 1.1 as well as the permanence of the equation. Also, we establish some