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 where a , b , A , B are positive real numbers, k ≥1 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 Kulenović et al. (2003) [The dynamics of x n +1 =(α+ βx n )/( A + Bx n + Cx n -1 ) facts and conjectures, Computational Mathematics Applications , 45 , 1087-1099]. § Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China. [ABSTRACT FROM AUTHOR]