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Abstract: In this paper the global asymptotic stability of the nonlinear difference equationis investigated, where α, β, A, B, C >0 are real numbers, and the initial conditions x −1 is nonnegative real numbers and x 0 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 facts and conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]. [Copyright &y& Elsevier]

In this paper, we investigate the periodic character, invariant intervals, oscillation and global stability of all positive solutions of the equationwhere the parameters p and q and the initial conditions x-k,…,x0 are nonnegative real numbers. As might be expected, the two cases p⩽q and p>q give rise to different dynamic behaviors. In particular, our results solve the open problem proposed by Kulenvic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations: with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2001]. [Copyright &y& Elsevier]

In this paper, we investigate the global stability of all negative solutions of the difference equationxn+1=α+xn−k/xn, n=0,1,…,where α<1 is a real number, k&ges;1 is an integer, and the initial conditions x−k,…,x0 are arbitrary real numbers. We show that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equationxn+1=a−bxn/A−∑ki=0bixn−i, n=0,1,…,where A,b,bk∈(0,∞), a,b0,…,bk−1∈[0,∞), k∈{1,2,…} and the initial conditions x−k,…,x0 are arbitrary real numbers. We show that one positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the periodic character, invariant intervals and the global attractivity of all positive solutions of the nonlinear difference equationxn+1=α+βxn/γ−xn−1, n=0,1,…,where α&ges;0, β,γ>0. We show that the positive equilibrium of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]

Our aim in this paper is to investigate the global attractivity of the recursive sequencexn+1=α−βxn/γ−xn−1, n=0,1,…,where α&ges;0, β,γ>0. We show that one positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients. [Copyright &y& Elsevier]