Results on Complex Valued Complete Fuzzy Metric Spaces

I. INTRODUCTION

In 1965, Zadeh [3] coined the term "fuzzy set." Following that, a slew of authors worked on fuzzy sets, expanding the fuzzy set theory and its applications [4-6]. The idea of fuzzy metric spaces was given by Kramosil and Michalik [7]. After then, George and Veeramani [9] updated this idea. Grabiec [8] investigated fuzzy metric space fixed-point theory. The idea of complex-valued metric spaces was introduced by Azam et al. [21].

Verma et al. [23] recently established 'Max' functions and the partial order relation for complex numbers, and used properties (E-A) and CLRg to prove fixed point theorems in complex valued metric space. Singh et al. [25] were the first to present the concept of complex-valued fuzzy metric spaces and to create the complex-valued fuzzy version of some metric space results.

The goal of this study is to expand well-known metric-space results to complex-valued fuzzy metric spaces and then prove them in complex-valued complete fuzzy metric spaces.

II. PRELIMINARIES

Def.2.1.[21]. Let be the set of complex numbers and , where . Then a partial order relation on is defined as follows:

Hence if one of the following satisfies;

In particular, if and one of (P02), (P03), and (P04) is satisfied, and we write if only (P04) is satisfied.

It can be noted that;

Def.2.2.[21]. Complex-Valued Metric Space (CVMS)

Let X be a non-empty set. Assume that the mappings satisfies:

(CV1) , for all and iff ;

(CV2) , for all ;

Then d is called a complex-valued metric on X, and is called a CVMS.

Def.2.3.[23]. The 'max' function with partial order relation is defined as

And the 'min' functions can be defined as

(1)

Following Zadeh's [3] contribution to fuzzy set theory, a number of scholars [4-6] contributed to the field's basics and core theories.

Buckley [10] was the first to present the concept of fuzzy complex numbers. Other authors were inspired by Buckley's work and continued their research on fuzzy complex numbers. Ramot et al. [1] expanded fuzzy sets to complex fuzzy sets in this chain.

Singh et al. [25], inspired by Ramot et al. [1,] constructed complex-valued fuzzy metric spaces using continuous t - norms, defined a Hausdorff topology on complex - valued fuzzy metric space, and gave the concept of Cauchy sequences in CVFMS.

We establish certain fixed-point conclusions in the situation of complex -valued fuzzy metric spaces, inspired by Singh et al. [25]. We begin by extending several well-known metric-space results to complex-valued fuzzy metric spaces, and then we prove those results in the setting of CVFMS.

Def.2.4.[1]. The complex fuzzy set is given by .

Where U is a universe of discourse, is a membership function and defined as where and both real-valued, with .

Def.2.5. [25]. Complex Valued Continuous t-norm

A binary operation , wherein and a fix , is called complex valued continuous t-norm if it satisfies the following conditions:

(1) * is associative and commutative,

(2) * is continuous,

(3)

(4) whenever and , for all , where .

Ex.2.5. [25]. The following binary operations defined in (i), (ii) and (iii) are complex valued continuous t-norm (i) .

(ii) , for a fix .

(iii) ; for a fix .

Def.2.6. [25]. Complex Valued Fuzzy Metric Spaces (CVFMS)

The triplet is said to be CVFMS if a complex valued fuzzy set M:

(where , * is a complex valued continuous t-norm) fulfil the following criteria:

(CF1)

(CF2) for all ,

(CF3) ,

(CF4)

(CF5) is continuous, for all and .

Note- Wherever appropriate to our study, we refer to [25] and the references mentioned in [25] for further basic definitions, examples, and fundamental features of CVMS.

Singh et al. [25] demonstrated the following lemmas in CVFMS before establishing the result on complex-valued fuzzy metric space, i.e. Theorem 2.7.

Lemma 2.7 [25]. Let be a CVFMS such that , for all , if , for all , , then .

Lemma 2.8 [25]. Let be a sequence in a CVFMS with , for all . If there exists a number which lies on such that

Then is a Cauchy sequence in .

The following theorem was established by Singh et al. [25], which is the resetting of the Banach contraction principle in CVFMS.

Theorem 2.7 [25]. Let be a CVFMS such that , and . Let be a mapping that satisfies , . Then has a fixed point that is unique.

III. MAIN RESULTS

Fisher [24] established the following theorem in complete metric space for three mappings.

Theorem A [24]. Let S and T be continuous mappings of a complete metric space (X, d) into themselves. Then S and T have a common fixed point in X iff a continuous mapping A of X into exists, which commutes with S and T and satisfies;

for all and . Indeed S, T and A have a unique common fixed point.

We can now extend the preceding theorem/result to complex-valued complete fuzzy metric space as follows:

Theorem -3.1. Let be a complex-valued complete fuzzy metric space (CVCFMS). S and T are continuous mappings from X to X. If A is a continuous mapping from X to , it commutes with S and T, and if detailed maps satisfy the following contractive condition.

for all and ... (3.11)

Additionally, , for all and ... (3.12)

Then S, T, and A have a unique common fixed point.

Proof: is a Cauchy sequence?

Since is a continuous mapping from to so for , there exists any such that and

On keep repeating this process for different and , we get a sequence such that

and

Or and

On setting and in (3.11), we get for r = 1, 2, 3,...

Now suppose

Then by (I), we have

By lemma (4.1) or (5.1), we have

Which is not possible

Hence by , we must have ... (II) In general, we get

Hence by lemma (4.2), is a Cauchy sequence in .

Since the space is complete, so there exists some such that

and

It follows that , and

On taking , then by lemma (4.1), we have;

This implies that

Thus p is a common fixed point of A, S, and T.

Uniqueness: - let be another fixed point of A, S, and T. Then, by (3.11), we have

Which implies that

As , and , also .

Then certainly we get,

Which implies that p = q.

As a result, p is unique.

Ex. 3.1. Let with the metric defined by .

For all and , we define or , , and -norm ' ' is defined as $a b = min{a,b}$ where , for and . Here, , for all .

is a CVCFMS with a given t-norm *.

are defined as:

And as:

The mappings S and T are continuous. A is continuous from X to .

Clearly, and

This implies that .

IV. CONCLUSION

Existing results from complete metric space have been extended to complex-valued complete fuzzy metric spaces in this study. We tested the extended version of the result using a new form of weaker contractive condition. We've offered an example that backup our major finding and proves our hypotheses. In this line, various complete metric space results can be extended and demonstrated in the context of complex-valued complete fuzzy metric spaces.