Global asymptotic stability of a second-order nonlinear difference equation

In this paper the global asymptotic stability of the nonlinear difference equationxn+1=α+βxnA+Bxn+Cxn-1,n=0,1,…is investigated, where α, β, A, B, C > 0 are real numbers, and the initial conditions x−1 is nonnegative real numbers and x0 is a positive real number. We show that the unique positive equilibrium of the equation is globally asymptotically stable. In particular, our results solve two conjectures proposed by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002] and by Kulenovic et al. in their paper [M.R.S. Kulenovic, G. Ladas, L.F. Martins, I.W. Rodrigues, The dynamics of xn+1=α+βxnA+Bxn+Cxn-1 facts and conjectures, Comput. Math. Appl. 45 (2003) 1087-1099].