Dynamics of higher order rational difference equation x_n+1 = (α + βxn)/(A + Bxn + Cx_n−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+1 = α+βxn/ a+bxn+cx n−k, n=0,1,2, ..., where the parameters α, β, a, b and c are positive, and the initial conditions x−k, x−k+1, . . . , x−1, x0 are positive real numbers and kɛ{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+1 = α+βxn / a+bxn+cx n-1 , facts and conjectures, comput. math. appl. 45 (2003) 1087–1099].