Global asymptotic stability for a higher order nonlinear rational difference equations

摘要

In this paper, we investigate the boundedness, periodic character, invariant intervals and the global asymptotic stability of all nonnegative solutions of the difference equationwhere 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].