MMN-3699

# Eventual periodicity of a max-type system of difference equations of higher order with four variables

**Guangwang Su**, College of Information and Statistics, Guangxi University, Finance and Economics, Nanning, 530003, China, and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce, Intelligent Information Processing, Nanning, 530003, China,`h198204c@163.com`

**Taixiang Sun**, Guangxi (ASEAN) Research Center of Finance and Economics, Nanning, 530003, China, and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing, Nanning, 530003, China,`q3009b@163.com`

**Caihong Han**, College of Information and Statistics, Guangxi Univresity, Finance and Economics, Nanning, 530003, China, and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce, Intelligent Information Processing, Nanning, 530003, China,`h198204c@163.com`

**Bin Qin**, Guangxi (ASEAN) Research Center of Finance and Economics, Nanning, 530003, China, and Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce, Intelligent Information Processing, Nanning, 530003, China,`q3009b@163.com`

**Weizhen Quan**, Zhanjiang Preschool Education College, Zhanjiang, 524300, China,`q3009b@163.com`

## Abstract

The aim of this paper is to investigate eventual periodicity of the following max-type system of difference equations of higher order with four variables
$$
\left\{\begin{array}{ll}u_{n} = \max\Big\{A ,\frac{s_{n-k}}{v_{n-1}}\Big\},\\
v_{n} = \max \Big\{B
,\frac{t_{n-k}}{u_{n-1}}\Big\},\\
s_{n} = \max\Big\{C ,\frac{u_{n-k}}{t_{n-1}}\Big\},\\
t_{n} = \max \Big\{D,\frac{v_{n-k}}{s_{n-1}}\Big\},\\
\end{array}\right. \ \ n\in
\{0,1,2,\cdots\}, $$ where $A,
B,C,D\in (0,+\infty)$ with $A\leq B$ and $C\leq D$, and the initial conditions
$u_{-i},v_{-i},s_{-i},t_{-i}\in (0,+\infty)$ for $i\in \{1,2,\cdots,k\}$. We show that: (1) If $AC< 1$ or $A=B=C=D=1$, then there exists a solution $\{(u_n,v_n,s_n,t_n)\}^{+\infty}_{n= -k}$ of this system which is not eventually periodic. (2)\ \ If $BD=AC= 1$ with $A\not=C$ or $BD>AC= 1$ or $AC> 1$, then every solution of this system is eventually periodic.

Vol. 23 (2022), No. 2, pp. 913-927

DOI: 10.18514/MMN.2022.3699