Asymptotic periodicity of a higher-order difference equation

Abstract

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn = f(x n-p1,...,xn-pk,xn-q1,...,xn-qm), n ∈ ℕ0, where pi, i ∈ {1,...,k}, and qj, j ∈ {1,...,m}, are natural numbers such that p1 < p2 <... < pk, q1 < q2 <... < qm and gcd(p1,...,pk,q1,...,qm) = 1, the function f ∈ C[(0,∞)k+m, (α, ∞)], α > 0, is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function g ∈ C[(α, ∞),(α, ∞)] such that g(g(x)) = x, x ∈ (α,∞), (equation), x ∈ (α, ∞), limx→a+g (x) = +∞, and lim x→+∞g(x) = α. It is proved that if all p i, i ∈{1,...,k}, are even and all qj, j ∈{1,...,m} are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.