Bounded implies eventually periodic for the positive case of reciprocal-max difference equation with periodic parameters

Abstract

This paper concerns the positive case of the difference equation with initial x terms positive, (A i ) n nonnegative, periodic in n and such that ∀ n ∃ i such that (A i ) n >0 We show that if a solution is bounded then it is eventually periodic. Previous results exist for k = 1 and 2. We first make a log transformation, replacing division by subtraction, then define a dynamical system that is equivalent to the difference equation. This system is shown to be nonexpansive in L ∞. A theorem by Weller, Hilbert's metric, part metric and selfmappings of a cone, PhD dissertation, Univ. of Bremin, West Germany, December 1987. (Available from the Center for Research Library, crl.edu), states that bounded solutions that are nonexpansive in a polyhedral norm, such as L ∞, have finite ω-limit sets. We prove that if a bounded solution has a finite ω-limit set then it must be eventually periodic. Therefore bounded implies eventually periodic for the log version. Finally, we apply this result to show that all positive solutions of the reciprocal difference equation with maximum are eventually periodic.