| MSC: | 39A10 | Difference equations, additive |
Abstract:
In this note we prove that every positive solution of the difference equation
\[
x_n=1+\frac{x_{n-k}}{x_{n-m}},\quad n=0,1,\ldots
\]
where $k,m\in {\bf N}$ are so that $k<m,$ and $2m=k(L+1)$ for some
$L\in {\bf N},$ converges to a $k$-periodic solution. A similar
result is proved for a corresponding symmetric system of
difference equations. We also consider the systems of difference
equations whose all solutions are periodic with the same period.
It is generalized and solved Open Problem 2.9.1 in
M.~R.~S.~Kulenovi\'c and G.~Ladas, {\it Dynamics of Second Order
Rational Difference Equations. With open problems and
conjectures}. Chapman and Hall/CRC, 2002.