Copyright © 2009 Sukanya Basu and Orlando Merino. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For nonnegative real numbers α, β, γ, A, B, and C such that B+C>0 and α+β+γ>0, the difference equation xn+1=(α+βxn+γxn−1)/(A+Bxn+Cxn−1), n=0,1,2,… has a unique positive equilibrium. A proof is given here for the following statements: (1) For every choice of positive parameters α, β, γ, A, B, and C, all solutions to the difference equation xn+1=(α+βxn+γxn−1)/(A+Bxn+Cxn−1), n=0,1,2,…,x−1,x0∈[0,∞) converge to the positive equilibrium or to a prime period-two solution. (2) For every choice of positive parameters α, β, γ, B, and C, all solutions to the difference equation xn+1=(α+βxn+γxn−1)/(Bxn+Cxn−1), n=0,1,2,…,  x−1,  x0∈(0,∞) converge to the positive equilibrium or to a prime period-two solution.