Global asymptotic stability of a second order rational difference equation

Abstract

The main goal of the paper is to confirm Conjecture 9.5.5 stated by Kulenovic and Ladas in Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures (Chapman & Hall/CRC, Boca Raton, FL, 2002). The boundedness, invariant intervals, semicycles and global attractivity of all nonnegative solutions of the equation χn+1 = βχn+γχn-1/A+Bχn+Cχ n-1, n∈ℕ0, are studied, where the parameters β, γ A, B, C ∈ (0, ∞) and the initial conditions χ-1,χ0 ∈ [0, ∞] are such that χ-1 + χ0 > 0. It is shown that if the equation has no prime period-two solutions, then the unique positive equilibrium of the equation is globally asymptotically stable.