GENERAL CYCLES OF POTENTIAL FORM

Abstract

We consider a general difference equation of order k of the form [Formula: see text] with f0 : (0, ∞) → (0, ∞), F : (0, ∞)k-1 → (0, ∞) continuous maps and initial conditions xi ∈ (0, ∞), i = 0, 1, …, k-1. Under the hypothesis that fx(y) := F(x, x, …, x)f0(y) is an involution, we provide all the (k + 1)-cycles of the above-mentioned type, and we prove that if F separates variables, then all the (k + 2)-cycles have a potential form [Formula: see text] where c > 0, αi ∈ ℝ, xi ∈ (0, ∞), i = 0, 1, …, k-1. For this reason, we also describe all the p-cycles of order k ≥ 2 having the above potential form. Moreover, taking the particular case k = 3 and p = 8 disproves Conjecture 2.1 from [Grove & Ladas, 2005].