## On the periodic character of some difference equations.(English)Zbl 1032.39004

The positive solutions $$\{ x_n\}_{n=-d}^\infty$$ of the difference equation $x_n=\max \left\{ \frac{A}{x_{n-k}},\frac{B}{x_{n-m}}\right\} \qquad n=0,1,\dots$ where $$k,m$$ are positive integers, $$d=\max\{ k,m\}$$ and $$A,B\in (0,\infty)$$, are investigated. It is shown that there exists a positive integer $$T$$ such that every positive solution $$\{ x_n\}_{n=-d}^\infty$$ is eventually periodic with period $$T$$, that is $x_{n+T}=x_n \qquad\text{for }n\geq n_0$ where $$n_0$$ may depend on the initial conditions $$x_{-1},x_{-2},\dots ,x_{-d}$$. In addition, the period $$T$$ is determined in terms of the parameters $$A$$, $$B$$, $$k$$, and $$m$$.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations and inequalities 39A10 Additive difference equations
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### References:

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