Document Type

Article

Date of Original Version

2009

Abstract

For nonnegative real numbers, a, β, y, A, B, and C such that B + C > 0 and a + β + y > 0, the difference equation xn+1 = (a + βxn + yxn-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 paramenters a, β, y, A, B, and C, all solutions to the difference equation xn+1 = (a, βxn, yxn-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 a, β, y, B, and C, all solutions to the difference equation xn+1 = (a, xn + yxn-1)/(Bxn + Cxn-1), n = 0, 1, 2, …, x-1, x0 ∈ (0,∞) converge to the positive equilibrium or to a prime period-two solution.

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Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

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