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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 x n-2

where {p n } n is positive and periodic with period k{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:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations