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Periodicity of a class of nonautonomous max-type difference equations. (English) Zbl 1225.39018

The paper contains new unifying periodicity criteria for the following difference equations
\[ x_n = \max\left\{f_1(x_{n-k_1},n),\ldots,f_m(x_{n-k_m},n),x_{n-s}\right\}\;,\;n\in{\mathbb N}_0 \]
with \(1\leq k_1<\dots<k_m\), \(m,s\in{\mathbb N}\), then for
\[ x_n = \min\left\{f_1(x_{n-k_1},n),\dots,f_m(x_{n-k_m},n),x_{n-s}\right\}\;,\;n\in{\mathbb N}_0 \]
with \(1\leq k_1<\dots<k_m\), \(m,s\in{\mathbb N}\) and finally for
\[ x_n=\max\left\{f_1(x_{n-k_1^{(1)}},\dots,x_{n-k_{t_1}^{(1)}}),\dots, f_m(x_{n-k_1^{(m)}},\dots,x_{n-k_{t_m}^{(m)}}),x_{n-s}\right\}\;,\;n\in{\mathbb N}_0 \]
with \(m,s\in{\mathbb N}\), \(t_i\in{\mathbb N}\), \(i=1,\dots,m\), \(1\leq k_1^{(i)}<\dots< k_{t_i}^{(i)}\), \(i=1,\dots,m\).
The assumptions are various monotonicity and periodicity conditions for the right hand side of the considered equations.

MSC:

39A23 Periodic solutions of difference equations
39A20 Multiplicative and other generalized difference equations
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References:

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