In this paper, we study the following max-type system of difference equations: $$\textstyle\begin{cases}x_{n} = \max \{A_{n},\frac{y_{n-1}}{x_{n-2}} \},\\ y_{n} = \max \{B_{n} ,\frac{x_{n-1}}{y_{n-2}} \}, \end{cases}\displaystyle n\in \{0,1,2,\ldots\}, $$ where \(A_{n},B_{n}\in(0, +\infty)\) are periodic sequences with period 2 and the initial values \(x_{-1},y_{-1},x_{-2},y_{-2}\in (0,+\infty)\). We show that every solution of the above system is eventually periodic.

We study in this paper the following max-type system of difference equations of higher order: xn=max{A,yn-k/xn-1} and yn=max{B,xn-k/yn-1}, n∈{0,1,2,…}, where A≥B>0, k≥1, and the initial conditions x-k,y-k,x-k+1,y-k+1,…,x-1,y-1∈(0,+∞). We show that (1) if AB>1, then every solution of the above system is periodic with period 2 eventually. (2) If AB=1>B, then every solution of the above system is periodic with period 2k or 2 eventually. (3) If A=B=1 or AB<1, then the above system has a solution which is not periodic eventually.

Let D be a dendrite and G be a finitely generated group. In this paper, we obtain some equivalent conditions of equicontinuous flow $(D,G)$ under some conditions.

In this paper, we study the eventual periodicity of the fuzzy max-type difference equation \(x_{n} =\max \{C , \frac{x_{n-m-k}}{x_{n-m} }\}, n\in \{0,1,\ldots \} \), where m and k are positive integers, C and the initial values are positive fuzzy numbers. Let the support \(\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]\) of C. We show that: (1) if \(C_{1}>1\), then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with \(C_{1}=1\) such that this equation has a positive solution which is not eventually periodic; (3) if \(C_{2}\leq 1\), then this equation has a positive solution which is not eventually periodic; (4) if \(C_{1}<1<C_{2}\), then every positive solution of the above equation is not eventually periodic.

Our aim in this paper is to investigate the convergence behavior of the positive solutions of a higher order fuzzy difference equation and show that all positive solutions of this equation converge to its unique positive equilibrium under appropriate assumptions. Furthermore, we give two examples to account for the applicability of the main result of this paper.