On the periodic nature of some max-type difference equations

Recently there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations. In [5, 6, 8] some global convergence results were established which can be applied to nonlinear difference equations in proving that every solution of these equations converges to a periodic solution (which need not necessarily be stable). The periodic nature of nonlinear difference equations of the max type has been investigated by many authors. See for example [1, 2, 3, 4]. Our main objective in this paper is to extend the study of boundedness and periodicity to solutions of some max-type difference equations. We deal with the following difference equation: