A note on periodic character of a difference equation

Abstract

In this note, we study positive solutions of the difference equation x n+1 = p + x n-(2s-1) /x n-(2l+1)s+1 , n = 0, 1... where p ∈ [1, ∞) and s, l ∈ N. We prove that if p > 1, then every positive solution converges to the positive equilibrium x* = p + 1 and if p = 1, then every positive solution converges to a 2s-periodic solution. The second result generalizes the main result in the paper: W.T. Patula and H.D.Voulov, On the oscillation and periodic character of a third order rational difference equation, Proc. Amer. Math. Soc. 131 (3) (2003), 905-909.