On the Dynamics of the Higher Order Nonlinear Rational Difference Equation

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k +Cxn−l +Dxn−σ + bxn−k [dxn−k − exn−l ] , n = 0,1,2, ....., where the coefficients A,B,C,D,b,d,e ∈ (0,∞), while k, l and σ are positive integers. The initial conditions x−σ ,...,x−l ,...,x−k, ...,x−1,x0 are arbitrary positive real numbers such that k < l < σ . Some numerical examples will be given to illustrate our results.