Boundedness character of a class of difference equations

Abstract

A complete picture regarding the boundedness character of positive solutions to the following difference equation xn = max {A, frac(underover(x, n - 1, p), underover(x, n - k, p))}, n ∈ N0, where k ≥ 2 and the parameters A and p are positive real numbers, is given. In particular, for the case pk - 1 ∈ (0, kk / (k - 1)k - 1), we prove that all solutions to the equation are bounded. We also present corresponding results concerning the following closely related difference equation xn = A + frac(xn - 1p, xn - kp), n ∈ N0 .