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.