Informal Complete Metric Space and Fixed Point Theorems

Hsien-Chung Wu

doi:10.3390/axioms8040126

Abstract

The concept of informal vector space is introduced in this paper. In informal vector space, the additive inverse element does not necessarily exist. The reason is that an element in informal vector space which subtracts itself cannot be a zero element. An informal vector space can also be endowed with a metric to define a so-called informal metric space. The completeness of informal metric space can be defined according to the similar concept of a Cauchy sequence. A new concept of fixed point and the related results are studied in informal complete metric space.

Highlights

The basic operations in vector space are vector addition and scalar multiplication.Based on these two operations, the vector space should satisfy some required conditions by referring to [1,2,3,4,5]
The space consisting of all fuzzy numbers in R cannot satisfy all of the axioms in vector space, where the addition and scalar multiplication of fuzzy sets are considered
In this paper, we propose the concept of null set for the purpose of playing the role of a zero element in the so-called informal vector space

Summary

Introduction

The basic operations in (conventional) vector space are vector addition and scalar multiplication. In this paper, we propose the concept of null set for the purpose of playing the role of a zero element in the so-called informal vector space. Based on the concept of null set, we can define the concept of almost identical elements in informal vector space. We can endow a metric to the informal vector space defining the so-called informal metric space This kind of metric is completely different from the conventional metric defined in vector space, since it involves the null set and almost identical concept. The main aim of this paper was to establish the so-called near-fixed point in informal, complete metric space, where the near fixed point is based on the almost identical concept.

Informal Vector Spaces

Informal Metric Spaces

Cauchy Sequences

Near Fixed Point Theorems