On some periodic systems of max-type difference equations. (English) Zbl 1280.39012
Summary: We show that all positive solutions to the system of max-type difference equations
\[ x_n^{(1)} = \max_{1\leq i \leq m_1} \left \{f_{1i} \left(x_{n-k_{i,1}^{(1)}}^{(1)},x_{n-k_{i,2}^{(1)}}^{(2)},\dots,x_{n-k_{i,l}^{(1)}}^{(l)},n \right),x_{n-s}^{(1)} \right\} , \]
\[ x_n^{(2)} = \max_{1\leq i \leq m_2} \left \{f_{2i} \left(x_{n-k_{i,1}^{(2)}}^{(1)},x_{n-k_{i,2}^{(2)}}^{(2)},\dots,x_{n-k_{i,l}^{(2)}}^{(l)},n \right),x_{n-s}^{(2)} \right\} , \]
\[ \vdots \]
\[ x_n^{(l)} = \max_{1\leq i \leq m_l} \left \{f_{li} \left(x_{n-k_{i,1}^{(l)}}^{(1)},x_{n-k_{i,2}^{(l)}}^{(2)},\dots,x_{n-k_{i,l}^{(l)}}^{(l)},n \right),x_{n-s}^{(l)} \right\} , \]
\(n \in \mathbb N_0\), where \(s,l,m_j,k_{i,t}^{(j)} \in \mathbb N\), \(t \in \{1,\dots,l\}\), and for a fixed \(j,i \in \{1,\dots,m_j\}\), and where the functions \(f_{ji} : (0,\infty)^l \times \mathbb N_0 \rightarrow (0,\infty)\), \(j \in \{1,\dots,l\}\), \(i \in \{1,\dots,m_j\}\) satisfy some conditions, are eventually periodic with (not necessarily prime) period \(s\). A related result for the corresponding system of min-type difference equations is also proved.
\[ x_n^{(1)} = \max_{1\leq i \leq m_1} \left \{f_{1i} \left(x_{n-k_{i,1}^{(1)}}^{(1)},x_{n-k_{i,2}^{(1)}}^{(2)},\dots,x_{n-k_{i,l}^{(1)}}^{(l)},n \right),x_{n-s}^{(1)} \right\} , \]
\[ x_n^{(2)} = \max_{1\leq i \leq m_2} \left \{f_{2i} \left(x_{n-k_{i,1}^{(2)}}^{(1)},x_{n-k_{i,2}^{(2)}}^{(2)},\dots,x_{n-k_{i,l}^{(2)}}^{(l)},n \right),x_{n-s}^{(2)} \right\} , \]
\[ \vdots \]
\[ x_n^{(l)} = \max_{1\leq i \leq m_l} \left \{f_{li} \left(x_{n-k_{i,1}^{(l)}}^{(1)},x_{n-k_{i,2}^{(l)}}^{(2)},\dots,x_{n-k_{i,l}^{(l)}}^{(l)},n \right),x_{n-s}^{(l)} \right\} , \]
\(n \in \mathbb N_0\), where \(s,l,m_j,k_{i,t}^{(j)} \in \mathbb N\), \(t \in \{1,\dots,l\}\), and for a fixed \(j,i \in \{1,\dots,m_j\}\), and where the functions \(f_{ji} : (0,\infty)^l \times \mathbb N_0 \rightarrow (0,\infty)\), \(j \in \{1,\dots,l\}\), \(i \in \{1,\dots,m_j\}\) satisfy some conditions, are eventually periodic with (not necessarily prime) period \(s\). A related result for the corresponding system of min-type difference equations is also proved.
MSC:
| 39A23 | Periodic solutions of difference equations |
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 39A10 | Additive difference equations |
Keywords:
system of max-type difference equations; system of min-type difference equations; eventually periodic solution; positive solutionReferences:
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