On the recursive sequence xn = 1 + ∑i=1 k αixn-pi/∑j=1 m βjxn-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=1k αi = ∑j=1m βj = 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.