Periodic character of a class of difference equation

Abstract

In this note, we prove that every positive solution of the difference equation x n+1 = p + qx n + qx n-k /x n-k +1, n = 0, 1... where k ∈ N, p, q ∈ [0, ∞) and the initial conditions x -k ,...,x -1 , x 0 are positive real numbers, converges to a periodic solution of period 2k. This result confirms conjectures 11.4.10, 11.4.16 (a) and 11.4.17 in Kulenović and Ladas (2001) Dynamics of Second Order Rational Difference Equations (Chapman and Hall/CRC).