The nature of solutions of the difference equation xn = max{A/xn-2,B/xn-3α}

We consider positive solutions of the difference equation x_n = max{A/x_n-2,B/xn-3α},n≥0, where A ≥ 0, B ≥ 0, 0 < α ≤ 1 and the initial conditions x_-1 x_-2, x_-3 are arbitrary positive real numbers. We show that every positive solution of this difference equation approaches x¯ = √B or is eventually periodic with period 4, 5 or 6. Also, this work confirms partially the conjecture proposed in [18].