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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}=\frac{\beta x_n+\gamma x_{n-1}}{A+Bx_n+Cx_{n-1}}, \quad n\in{\mathbb N}_0, \tag{\(*\)} \] where \(\beta,\gamma,A,B,C\in(0,\infty)\) and the initial conditions \(x_{-1},x_0\in[0,\infty)\) 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.

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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