Stability of the recursive sequence \(x_{n+1}=(\alpha-\beta x_n)/(\gamma+x_{n-1})\). (English) Zbl 0990.39009

Consider the recursive sequence \[ x_{n+1}= {\alpha+ \beta x_n \over \gamma+x_{n-1}},\;n=0,1,\dots \tag{*} \] where \(\alpha,\beta\) and \(\gamma\) are nonnegative and the initial conditions \(x_1\) and \(x_0\) are arbitrary. Equation (*) has two equilibrium points positive and negative.
If there exists \(k\geq 2\) such that \(\gamma\geq k\alpha/ \beta\) and \(\alpha\geq k\beta^2\), then the positive equilibrium point is a global attractor with some given basin. The asymptotic properties in the case \(\alpha=0\), \(\beta<0\), \(\gamma >0\) are investigated in details.


39A11 Stability of difference equations (MSC2000)
39B05 General theory of functional equations and inequalities
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