Dynamics of a higher order nonlinear rational difference equation

In this paper, we study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equation x(n+1) = a+bx(n)/Ax(n) + Bx(n-k), n = 0,1..., where a , b , A , B are positive real numbers, k greater than or equal to1 is a positive integer, and the initial conditions x(-k) ,..., x(-1) , x(0) are nonnegative real numbers such that x(-k) or x(0) 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 Kulenovic et al. (2003) [The dynamics of x(n+1) =(alpha+ betax(n))/( A + Bx(n) + Cx(n-1)) facts and conjectures, Computational Mathematics Applications , 45 , 1087-1099].