We investigate the global asymptotic stability of the solutions of X n+1 = βX n−l +γX n−k A+X n−k 2010 Mathematics Subject Classification. 39A10, 39A11.

We investigate the global character of solutions of the difference equation in the title with nonnegative parameters, nonnegative initial conditions, and where k is a positive integer. In particular, for the case k = 2 we will address the Open Problem 5.173.1 posed in E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2008.

Dedicated to Professor Ronald E. Mickens on the occasion of his 60th Birthday. May his outstanding contributions continue for many years.We investigate the global stability, the periodic character, and the boundedness nature of the solutions of the difference equationwhere the parameters a, g, A, B and the initial conditions x 22 , x 21 , x 0 are non-negative real numbers. We show that the solutions of the equation exhibit a trichotomy character which depends upon whether g is less than A, equal to A, or greater than A.

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