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

Miskolc Mathematical Notes

article

Guangwang Su¹  Taixiang Sun³  Caihong Han¹  Bin Qin³  Weizhen Quan⁵ 

[1] College of Information and Statistics;Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce;Guangxi ,(ASEAN) Research Center of Finance and Economics;Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing;Zhanjiang Preschool Education College

关键词: Max-type system;    Difference equation;    Positive solution;    Higher order;    Eventual periodicity;   

DOI  :  10.18514/MMN.2022.3699

学科分类：数学（综合）

来源: Miskolci Egyetem

【 摘 要 】

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 $AC 1$, then every solution of this system is eventually periodic.

【 授权许可】

CC BY