Advances in Difference Equations
Volume 2009 (2009), Article ID 128602, 27 pages
doi:10.1155/2009/128602

Global behavior of solutions to two classes of second-order rational difference equations

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.