Research Article On Global Periodicity of a Class of Difference Equations

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}.