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A rational difference equation. (English) Zbl 0985.39017

The authors investigate the nonlinear rational difference equation \[ x_{n+1}= (\alpha x_n+ \beta x_{n-1})/(A+x_{n-1}),\;n=0,1,2,\dots, \tag{*} \] where the parameters \(\alpha,\beta\) and \(A\) and the initial conditions \(x_{-1}\) and \(x_0\) are nonnegative real numbers. The Riccati difference equation and the Pielou discrete delay logistic model are obtained from (*) for \(\alpha=0\) and \(\beta=0\), respectively. The boundedness character, the periodic nature, and the global asymptotic stability of all positive solutions of the equation (*) with \(\alpha> 0\), \(\beta>0\), \(A\geq 0\), are studied.

MSC:

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

[1] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0787.39001
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