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Zbl 1103.39004
Berg, Lothar; Stević, Stevo
Periodicity of some classes of holomorphic difference equations. (English)
[J] J. Difference Equ. Appl. 12, No. 8, 827-835 (2006). ISSN 1023-6198

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.
[Vladimir Răsvan (Craiova)]

MSC 2000:: *39A11 Stability of difference equations
39A20 Generalized difference equations

Keywords: rational difference equation; $p$-periodic solution; holomorphic solution; stability; unbounded solutions; positive equilibrium

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