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

Abstract

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

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