Global Behavior of Solutions to Two Classes of Second Order Rational Difference Equations

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