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Zbl 1177.39016
Basu, Sukanya; Merino, Orlando
Global behavior of solutions to two classes of second-order rational difference equations.
(English)
[J] Adv. Difference Equ. 2009, Article ID 128602, 27 p. (2009). ISSN 1687-1847/e

Summary: For nonnegative real numbers $\alpha,\beta,\gamma, A, B$, and $C$ such that $B+C>0$ and $\alpha+\beta+\gamma>0$, the difference equation $$x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{A+Bx_n+Cx_{n-1}},\quad n=0,1,2,\dots$$ has a unique positive equilibrium. A proof is given here for the following statements: (1) For every choice of positive parameters $\alpha,\beta,\gamma, A, B$, and $C$, all solutions to the difference equation $$x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{A+B x_n+Cx_{n-1}},\quad n=0,1,2,\dots,x-1,\ x_0\in [0,\infty)$$ converge to the positive equilibrium or to a prime period-two solution. (2) For every choice of positive parameters $\alpha,\beta,\gamma$, $B$, and $C$, all solutions to the difference equation $$x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}}{B x_n+C x_{n-1}}, \quad n=0,1,2,\dots,x_{-1},\ x_0\in(0,\infty)$$ converge to the positive equilibrium or to a prime period-two solution.
MSC 2000:
*39A23
39A20 Generalized difference equations

Keywords: second-order rational difference equations; positive equilibrium; period-two solution

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