On the boundedness of positive solutions of the reciprocal max-type difference equation x_n = max A_(n-1) ~1/x _(n-1), A_(n-1) ~2/x_(n-2), ?, A _(n-1) ~t x_(n-t) with periodic parameters

摘要

We investigate the boundedness of positive solutions of the reciprocal max-type difference equation xn = max A_(n-1) ~1/x_(n-1), A_(n-1) ~2/x_(n-2), ?, A_(n-1) ~t x_(n-t),n=1,2,?,where, for each value of i, the sequence An i n=0 of positive numbers is periodic with period pi. We give both sufficient conditions on the pi's for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. This work essentially complements the work by Bidwell and Franke, who showed that as long as every positive solution of our equation is bounded, then every positive solution is eventually periodic, thereby leaving open the question as to when solutions are bounded.