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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.

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