Dynamics of a higher order nonlinear rational difference equation

Abstract

In this paper, we study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equation x n+1 =a + bxn/Axn+Bxn-k, n= 0, 1,..., where a, b, A, B are positive real numbers, k ≥ 1 is a positive integer, and the initial conditions x-k,...,x-1,-x0 are nonnegative real numbers such that x-k or x0 or both are positive real numbers. We show that the positive equilibrium of the difference equation is a global attractor. As a corollary, our main result confirms a conjecture proposed by Kulenović et al. (2003) [The dynamics of xn+1 = (α + βxn)/(A + Bxn + Cxn-1) facts and conjectures, Computational Mathematics Applications, 45, 1087-1099].