On the Recursive Sequence xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj

Abstract

We give a complete picture regarding the behavior of positive solutions of the following important difference equation: xn=1+∑i=1kαixn−pi/∑j=1mβjxn−qj, n∈ℕ0, where αi, i∈{1,…,k}, and βj, j∈{1,…,m}, are positive numbers such that ∑i=1kαi=∑j=1mβj=1, and pi, i∈{1,…,k}, and qj, j∈{1,…,m}, are natural numbers such that p1<p2<⋯<pk and q1<q2<⋯<qm. The case when gcd(p1,…,pk,q1,…,qm)=1 is the most important. For the case we prove that if all pi, i∈{1,…,k}, are even and all qj, j∈{1,…,m}, are odd, then every positive solution of this equation converges to a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.