Global asymptotical stability of a second order rational difference equation

摘要

In this paper, we investigate the boundedness, invariant interval, semicycle and global attractivity of all positive solutions of the equation x_(n+1) = α+γx_ (n-1)/A+Bx_n+Cx_(n-1) ,n= 0, 1,..., where the parameters α, γ, A, B, C ∈ (0, ∞) and the initial conditions y-1, y_0 are nonnegative real numbers. We show that if the equation has no prime period-two solutions, then the positive equilibrium of the equation is globally asymptotically stable. Our results solve partially the conjecture proposed by Kulenovic and Ladas in their monograph [M.R. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001].