On global periodicity of a class of difference equations

Abstract

We show that the difference equation xn = f3 (xn-1) f2 (xn-2) f1 (xn-3), n ∈ ℕ0, where fi ∈ C [(0, ∞), (0, ∞)], i ∈ {1, 2, 3}, is periodic with period 4 if and only if fi (x) = ci/x for some positive constants ci, i ∈ {1, 2, 3} or if fi (x) = ci/x when i = 2 and fi (x) = cix if i ∈ {1, 3}, with c1c2c3 = 1. Also, we prove that the difference equation xn = f4 (xn-1) f3 (xn-2) f2 (xn-3) f1 (xn-4), n ∈ ℕ0, where fi ∈ C [(0, ∞), (0, ∞)], i ∈ {1, 2, 3, 4}, is periodic with period 5 if and only if fi (x) = ci/x, for some positive constants ci, i ∈ {1, 2, 3, 4}.