Boundedness character of a class of difference equations

A complete picture regarding the boundedness character of positive solutions to the following difference equation xn=max{A,xn−1pxn−kp},n∈N0, where k≥2k≥2 and the parameters AA and pp are positive real numbers, is given. In particular, for the case pk−1∈(0,kk/(k−1)k−1)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+xn−1pxn−kp,n∈N0.