On the Recursive Sequencexn+1=A+xnp/xn−1

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<4r, or 2r≤p<1+r, r∈(0,1), then all positive solutions of the equation are bounded. Also, an analogous result is provedregarding positive solutions of the max type difference equation xn+1=max{A,xnp/xn−1r}, where A, p, q∈(0,∞)