Stevo Stevi\' c
Periodic Character of a Difference Equation
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 59, 3-10(2005)

MSC:

39A10 Difference equations, See also {33Dxx}

39A11 Stability of difference equations

Abstract: In this note we prove that every positive solution of the difference equation

$$x_{n+1}=\frac{x_{n-1}}{p+x_{n-1}+x_n},\quad n=0,1...$$

where $p\in [0,\infty)$ and the initial conditions $x_{-1},x_0$ are positive real numbers, converges to a, not necessarily prime,

periodic-two solution.

This result confirms Conjecture 7.5.2 in \cite{ste-l} (with $q=1$). Also, we show that the positive solutions of Eq.(1) converge to

the corresponding periodic-two solutions geometrically.
Keywords: Period two solution, difference equation, positive solution, asymptotics