an:05057372
Zbl 1103.39004
Berg, Lothar; Stevi??, Stevo
Periodicity of some classes of holomorphic difference equations
EN
J. Difference Equ. Appl. 12, No. 8, 827-835 (2006).
00185385
2006
j
39A11 39A20
rational difference equation; \(p\)-periodic solution; holomorphic solution; stability; unbounded solutions; positive equilibrium
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 R??svan (Craiova)