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(xn−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,∞), x=f(x,…,x︸k,g(x),…,g(x)︸m), x∈(α,∞), limx→α+g(x)=+∞, and limx→+∞g(x)=α. It is proved that if all pi, 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.