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Near Fixed Point Theorems in the Space of Fuzzy Numbers

Author

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  • Hsien-Chung Wu

    (Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan)

Abstract

The fuzzy numbers are fuzzy sets owning some elegant mathematical structures. The space consisting of all fuzzy numbers cannot form a vector space because it lacks the concept of the additive inverse element. In other words, the space of fuzzy numbers cannot be a normed space even though the normed structure can be defined on this space. This also says that the fixed point theorems established in the normed space cannot apply directly to the space of fuzzy numbers. The purpose of this paper is to propose the concept of near fixed point in the space of fuzzy numbers and to study its existence. In order to consider the contraction of fuzzy-number-valued function, the concepts of near metric space and near normed space of fuzzy numbers are proposed based on the almost identical concept. The concepts of Cauchy sequences in near metric space and near normed space of fuzzy numbers are also proposed. Under these settings, the existence of near fixed points of fuzzy-number-valued contraction function in complete near metric space and near Banach space of fuzzy numbers are established.

Suggested Citation

  • Hsien-Chung Wu, 2018. "Near Fixed Point Theorems in the Space of Fuzzy Numbers," Mathematics, MDPI, vol. 6(7), pages 1-27, June.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:7:p:108-:d:154293
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    References listed on IDEAS

    as
    1. Cho, Yeol Je & Sedghi, Shaban & Shobe, Nabi, 2009. "Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2233-2244.
    2. Abu-Donia, H.M., 2007. "Common fixed point theorems for fuzzy mappings in metric space under ϕ-contraction condition," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 538-543.
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