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On positive solutions of a system of max-type difference equations. (English) Zbl 1293.39007

The authors consider the system \[ \displaystyle{x_{n+1} = \max\left\{c,{{y_n^p}\over{z_{n-1}^p}}\right\}\;,\;y_{n+1} = \max\left\{c,{{z_n^p}\over{x_{n-1}^p}}\right\}\;,\;z_{n+1} = \max\left\{c,{{x_n^p}\over{y_{n-1}^p}}\right\}} \] with \(p\), \(c\) positive reals.
A solution is called positive if \(\min\{x_n,y_n,z_n\}>0\), \(\forall n\geq -1\). If there exist \(m\), \(M\) such that \[ m\leq\min\{x_n,y_n,z_n\}\leq M \] holds for sufficiently large \(m\geq -1\), the system is called permanent. The following statements are proved:
1. If \(p\geq 4\) and \(c>0\), there exist positive bounded solutions.
2. If \(p\in(0,4)\) and \(c>0\), the system is permanent.
3. If \(p\in(0,4)\) and \(c\geq 1\), then every positive solution is eventually equal to \((c,c,c)\).
4. If \(p\in(0,1)\) and \(c\in (0,1)\), then every positive solution converges to \((1,1,1)\).

MSC:

39A20 Multiplicative and other generalized difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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