The Dynamics of the Solutions of Some Difference Equations

This paper is devoted to investigate the global behavior of the following rational difference equation: yn 1 αyn−t/ β γ ∑k i 0 y p n− 2i 1 ∏k i 0y 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 x0, x−1, . . . , x−2k, x−2max{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.