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