Dynamics of higher order rational difference equation xn+1=(α+βxn)/(A+Bxn+Cxn−k)

The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation
xn=αβxnAܙ︋−k, n=0,1,2,…,
where the parameters α, β, A, B and C are positive, and the initial conditions x−k,x−k,…,x−1,x0 are positive real numbers and k∈{1,2,3,…}. We give a detailed description of the semi-cycles of solutions and determine conditions under which the equilibrium points are globally asymptotically stable. In particular, our paper is a generalization of the rational difference equation that was investigated by Kulenovic et al. [The Dynamics of xn=αβxnAܙ︋−1
, Facts and Conjectures, Comput. Math. Appl. 45 (2003) 1087--1099].