On ω-Limit Sets of Zadeh’s Extension of Nonautonomous Discrete Systems on an Interval

Mathematics

Taixiang Sun¹  Guangwang Su¹ 

[1] College of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, China;

关键词: fuzzy number;    ω-limit set;    periodic point;    zadeh’s extension;   

DOI  :  10.3390/math7111116

来源: DOAJ

【 摘 要 】

Let $I=[0,1]$ and $f_n$ be a sequence of continuous self-maps on I which converge uniformly to a self-map f on I. Denote by $\mathcal{F}\left(I\right)$ the set of fuzzy numbers on I, and denote by $(\mathcal{F}\left(I\right),\widehat{f})$ and $(\mathcal{F}\left(I\right),{\widehat{f}}_n)$ the Zadeh ${}^{\prime }$ s extensions of $(I,f)$ and $(I,f_n)$ , respectively. In this paper, we study the $\omega$ -limit sets of $(\mathcal{F}\left(I\right),{\widehat{f}}_n)$ and show that, if all periodic points of f are fixed points, then $\omega (A,{\widehat{f}}_n)\subset F\left(\widehat{f}\right)$ for any $A\in \mathcal{F}\left(I\right)$ , where $\omega (A,{\widehat{f}}_n)$ is the $\omega$ -limit set of A under $(\mathcal{F}\left(I\right),{\widehat{f}}_n)$ and $F\left(\widehat{f}\right)=\{A\in \mathcal{F}\left(I\right):\widehat{f}\left(A\right)=A\}$ .

【 授权许可】

Unknown