Asymptotic Periodicity of a Higher-Order Difference Equation

We give a complete picture regarding the asymptotic periodicity of positivesolutions 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 p10, isincreasing in the first k arguments and decreasing in other marguments, there is a decreasing function g∈C[(α,∞),(α,∞)] such that g(g(x))=x, x∈(α,∞),x=f(x,…,xk,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