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On second-order rational difference equations. I. (English) Zbl 1131.39005
The authors present a summary of recent results concerning the global behaviour of the positive solutions of the rational difference equation \[ x_{n+1}= \Biggl(\alpha+ \sum^k_{i=1} \beta_i x_{n-i}\Biggr)\left/\Biggl(A+ \sum^k_{i=1} B_i x_{n-i}\Biggr)\right. \] with nonnegative coefficients together with some new results, open problems and conjectures. Emphasis is laid on boundedness, global stability of the equilibrium, periodicity, and convergence to periodic solutions, and trichotomies, in particular, in the case \(k= 2\), but also concerning some more general equations.

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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