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

Authors

Sukanya Basu, University of Rhode Island
Orlando Merino, University of Rhode IslandFollow

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 x_n+1 = (a + βx_n + yx_n-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 x_n+1 = (a, βx_n, yx_n-1)/(A + Bx_n + Cx_n-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 x_n+1 = (a, x_n + yx_n-1)/(Bx_n + Cx_n-1), n = 0, 1, 2, …, x_-1, x₀ ∈ (0,∞) converge to the positive equilibrium or to a prime period-two solution.

Citation/Publisher Attribution

Basu, S., & Merino, O. (2009). Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations. Advances in Difference Equations, 2009, Article ID: 128602. doi: 10.1155/2009/128602

Available at: https://doi.org/10.1155/2009/128602

Creative Commons License

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