Near Fixed Point Theorems in the Space of Fuzzy Numbers

Hsien-Chung Wu

doi:10.3390/math6070108

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.

Highlights

The fuzzy numbers can be treated as the imprecise data
The purpose of this paper is to study the near fixed point theorem in the near metric space Fcc (R)
The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach space of fuzzy numbers

Summary

Introduction

The fuzzy numbers can be treated as the imprecise data. For example, in the financial market, the data may not be precisely measured owing to the fluctuation. The only difference is that the concept of null set is involved in the axioms Under these settings, we shall study the so-called near fixed point theorem in the near normed space of fuzzy numbers (Fcc (R), k · k). Since the null set Ω can play the role of “zero element” in Fcc (R), i.e., the elements ω (1) and ω (2) can be ignored in some sense, e this kind of fuzzy number ã is said to be a near fixed point of the mapping T. In this case, many near fixed point theorems in near Banach space of fuzzy numbers are established

Space of Fuzzy Numbers

Near Metric Space of Fuzzy Numbers

Near Normed Space of Fuzzy Numbers

Cauchy Sequences in Near Metric Space of Fuzzy Numbers

Cauchy Sequences in Near Normed Space of Fuzzy Numbers

Near Fixed Point Theorems in Near Banach Space of Fuzzy Numbers