Global asymptotical stability of a second order rational difference equation

In this paper, we investigate the boundedness, invariant interval, semicycle and global attractivity of all positive solutions of the equation xn+1=α+γxn−1A+Bxn+Cxn−1,n=0,1,…, where the parameters α,γ,A,B,C∈(0,∞)α,γ,A,B,C∈(0,∞) and the initial conditions y−1,y0y−1,y0 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 Kulenović and Ladas in their monograph [M.R. Kulenović, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman Hall/CRC, Boca Raton, 2001].