On the periodic nature of the solutions of the reciprocal difference equation with maximum

We prove that every positive solution of the third order difference equation xn = max A xn−1 , B xn−2 , C xn−3 , n= 0, 1, . . . , is eventually periodic of (not necessarily prime) period T , which is explicitly determined by the coefficients A, B, and C.  2004 Elsevier Inc. All rights reserved.