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Global behavior of solutions to two classes of second-order rational difference equations. (English) Zbl 1177.39016
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:
39A23 Periodic solutions of difference equations
39A20 Multiplicative and other generalized difference equations
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References:
[2] doi:10.1080/10236199508808020 · Zbl 0858.39002
[4] doi:10.1016/S0893-9659(00)00068-9 · Zbl 0958.39021
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