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Pointwise equicontinuity of Zadeh’s extension of an interval map

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  • Sun, Taixiang
  • Su, Guangwang
  • Qin, Bin

Abstract

Let I be a compact interval and f:I⟶I be continuous. Assume that F(I) is the set of fuzzy numbers on I, and that f^:F(I)⟶F(I) is the Zadeh′s extension of f, and that lim←{F(I),f^} is the inverse limit space of (F(I),f^), and σf^:lim←{F(I),f^}⟶lim←{F(I),f^} is the left shift map. In this paper, we study the pointwise equicontinuity of f^ and show that the following statements are equivalent: (1) f^ is pointwise equicontinuous. (2) For some infinite subsequence S of positive integers, f^ is S pointwise equicontinuous. (3) {f^2n}n=1∞ is uniformly convergent on F(I). (4) σf^ is a periodic map with period 2.

Suggested Citation

  • Sun, Taixiang & Su, Guangwang & Qin, Bin, 2019. "Pointwise equicontinuity of Zadeh’s extension of an interval map," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 1-4.
    • Handle: RePEc:eee:chsofr:v:128:y:2019:i:c:p:1-4
    DOI: 10.1016/j.chaos.2019.07.031
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    References listed on IDEAS

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    1. Sun, Taixiang & Chen, Zhanhe & Liu, Xinhe & Xi, Hongjian, 2014. "Equicontinuity of dendrite maps," Chaos, Solitons & Fractals, Elsevier, vol. 69(C), pages 10-13.
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