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Global asymptotic stability of a second order rational difference equation. (English) Zbl 1153.39015

This paper studies the properties of solutions of the rational difference equation

x n+1 =βx n +γx n-1 A+Bx n +Cx n-1 ,n 0 ,(*)

where β,γ,A,B,C(0,) and the initial conditions x -1 ,x 0 [0,) are not both zero.

The main result answers positively the open problem posed by M. R. S. Kulenović and G. Ladas [Dynamics of second-order rational difference equations. With open problems and conjectures, Boca Raton, FL: Chapman & Hall/CRC (2002; Zbl 0981.39011), Conjecture 9.5.5], i.e., the positive equilibrium point of equation (*) is globally asymptotically stable. Furthermore, the authors prove the boundedness of every nonnegative solution and provide a detailed analysis of the invariant intervals and semicycles.

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations