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

Abstract

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 + Cxn−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 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+1 = α+βxn A+Bxn+Cxn−1 , Facts and Conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]