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Zbl 1153.39015
Hu, Lin-Xia; Li, Wan-Tong; Stević, Stevo
Global asymptotic stability of a second order rational difference equation.
(English)
[J] J. Difference Equ. Appl. 14, No. 8, 779-797 (2008). ISSN 1023-6198

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. \par The main result answers positively the open problem posed by {\it M. R. S. Kulenović} and {\it 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.
[Roman Šimon Hilscher (Brno)]
MSC 2000:
*39A11 Stability of difference equations
39A20 Generalized difference equations

Keywords: bounded solution; invariant interval; semicycle; global attractor; asymptotic stability; rational difference equation; positive equilibrium; nonnegative solution

Citations: Zbl 0981.39011

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