Eventual periodicity of the fuzzy max-difference equation x n = max { C , x n − m − k x n − m } $x_{n} = \max \{ C, \frac{x_{n-m-k}}{x_{n-m}}\}$

Abstract In this paper, we study the eventual periodicity of the fuzzy max-type difference equation x n = max { C , x n − m − k x n − m } , n ∈ { 0 , 1 , … } $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 supp C = { t : C ( t ) > 0 } ‾ = [ C 1 , C 2 ] $\operatorname{supp} C=\overline{\{t : C(t) > 0\}}=[C_{1},C_{2}]$ of C. We show that: (1) if C 1 > 1 $C_{1}>1$ , then every positive solution of this equation equals C eventually; (2) there exists a positive fuzzy number C with C 1 = 1 $C_{1}=1$ such that this equation has a positive solution which is not eventually periodic; (3) if C 2 ≤ 1 $C_{2}\leq 1$ , then this equation has a positive solution which is not eventually periodic; (4) if C 1 < 1 < C 2 $C_{1}