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}.