## On the recursive sequence $$x_{n+1}=\frac{\alpha+\beta x_{n-k}}{f(x_n,\dots,x_{n-k+1})}$$.(English)Zbl 1100.39014

The paper discusses qualitative properties for the solutions of the difference equation $x_{n+1}= {{(\alpha+\beta x_{n-k})}\over{f(x_n,\ldots,x_{n-k+1})}}$ with $$\alpha\geq 0\;,\;\beta\geq 0$$ and $$f:\mathbb R_+^k\to\mathbb R_+$$ continuous and non-decreasing in each argument such that $$f(0,0,\ldots,0)>0$$. Only nonnegative solutions are considered. Several cases are tackled: $$\beta<1$$; $$\beta>1$$; $$\beta=1, \alpha>0$$. An open problem is finally stated.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
Full Text: