Global asymptotic stability for a higher order nonlinear rational difference equations

In this paper, we investigate the boundedness, periodic character, invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equationxn+1=axn+bxn-kA+Bxn,n=0,1,…where a, b, A, B are positive real numbers, k ⩾ 1 is a positive integer, and the initial conditions x−k, … , x−1, x0 are nonnegative real numbers. It is shown that the zero equilibrium of this equation is globally asymptotically stable if b ⩽ A − a and the unique positive equilibrium is globally asymptotically stable if A − a < b < A + a. The results obtained solve a open problem proposed by Kulenovic and Ladas [M.R.S. Kulenovi, G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, FL, 2002, p. 129].