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Periodicity of some classes of holomorphic difference equations. (English) Zbl 1103.39004
The authors consider the difference equation \[ x_{n+1}=p_n+{{x_{n-1}}\over{x_{n-2}}} \] where \(\{p_n\}_n\) is positive and periodic with period \(k\in\{2,3\}\). The initial conditions are positive. In the case \(k=2\) it is proved that there are no solutions of odd period; then stability by the first approximation of the equilibrium is considered. Further global results are given for an associated system of three difference equations. This will lead to a global stability result for the basic equation.
Next, sufficient conditions for the existence of unbounded solutions are given. In the case \(k=3\) the following results are obtained: existence of a unique positive equilibrium using such classical results as Theorems of Descartes and Rolle; this equilibrium is stable by the first approximation. Existence of unbounded solutions is obtained also in this case.

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