On the recursive sequence Xn+1= A + xnp/ xn-1r

Abstract

The paper considers the boundedness character of positive solutions of the difference equation xn+1= A + xnp/ xn-1r, n∈ ℕ0, where A, p, and r are positive real numbers. It is shown that (a) If p2 ≥ 4r > 4, or p ≥ 1 + r, r ≤ 1, then this equation has positive unbounded solutions; (b) if p2 < 4 r, or 2 √r ≤ p < 1 + r, r ∈ (0, 1), then all positive solutions of the equation are bounded. Also, an analogous result is proved regarding positive solutions of the max type difference equation xn+1 = max {A, xnp/xn-1r}, where A, p, q ∈ (0, ∞).