Dynamics of some Rational Difference Equations

The main goal of this paper is to investigate the qualitative behavior of the solutions for the following rational difference equation: x(n+1) alpha + Sigma(k)(i=0) alpha(2i)x(n-2i)/beta+Sigma(k)(i=0) b(2i+1)x(n-2i-1) , n = 0, 1, 2, ... where alpha, beta, a(i), b(i) is an element of (0, infinity), i = 0,1,..., k; with the initial conditions x(0), x(-1), ..., x-(2k), x(-2k-1) is an element of (0, infinity). We determine the equilibrium points of the considered equation and then study their local stability. Also we study the boundedness and the permanence of the solutions. Finally we investigate the global asymptotically stable of the equilibrium points.