In this paper, we study the global attractivity, the invariant intervals, the periodic and oscillatory character of the difference equationwhere 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 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.