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][5][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.

Def.2.1. [21]. Let ? be the set of complex numbers and ? 1 , ? 2 ? ?, where ? = ? + ??. Then a partial order relation '? ' on ? is defined as follows:

Hence ? 1 ? ? 2 if one of the following satisfies;

London Journal of Research in Science: Natural and Formal (PO1) ??(? 1 ) = ??(? 2 ) and ??(? 1 ) = ??(? 2 ) (PO2) ??(? 1 ) < ??(? 2 ) and ??(? 1 ) = ??(? 2 ) (PO3) ??(? 1 ) = ??(? 2 ) and ??(? 1 ) < ??(? 2 ) (PO4) ??(? 1 ) < ??(? 2 ) and ??(? 1 ) < ??(? 2 )

In particular, ? 1 ? ? 2 if ? 1 ? ? 2 and one of (PO2), (PO3), and (PO4) is satisfied, and we write ? 1 ? ? 2 if only (PO4) is satisfied.

It can be noted that;

Let ? be a non-empty set. Assume that the mappings ?: ? × ? ? ? satisfies: (CV1) 0 ? ?(?, ?), for all ?, ? ? ? and ?(?, ?) = 0 iff ? = ? ;

(CV2) ?(?, ?) = ?(?, ?), for all ?, ? ? ? ;

(CV3) ?(?, ?) ? ?(?, ?) + ?(?, ?), for all ?, ?, ? ? ? Then ? is called a complex-valued metric on ?, 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

Following Zadeh's [3] contribution to fuzzy set theory, a number of scholars [4][5][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.

] 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 ? is a universe of discourse, ? ? (?) is a membership function and defined as ? ? (?) = ? ? (?). ? ?? ? (?) The triplet (?, ?, * ) is said to be CVFMS if a complex valued fuzzy set ? ? ? × ? × (0, ?) ? ? ? ? ?? (where ? ? ?, * is a complex valued continuous t-norm) fulfil the following criteria: and ? > 0. Let ?: ? ? ? be a mapping that satisfies ?(??, ??, ??) ? ?(?, ?, ?), ? ? ? (0, 1). Then ? has a fixed point that is unique.

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 S(X) ?T(X) exists, which commutes with S and T and satisfies;

?(??, ??) ? ? ?(??, ??) for all ?, ? ? ? and 0 < ? < 1. Indeed ?, ? and ? 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). ? and ? are continuous mappings from ? to ?. If ? is a continuous mapping from ? to ?(?) ? ?(?), it commutes with ? and ?, and if detailed maps satisfy the following contractive condition.

?(??, ??, ??) ? ???{ ?(??, ??, ?), ?(??, ??, ?), ?(??, ??, ?)} for all ?, ? ? ?, ? ? (0, ?) and 0 < ? < 1 ? (3.11)

Additionally, lim ??? ?(?, ?, ?) = ? ?? , for all ?, ? ? ? and ? ? [0, Then ?, ?, and ? have a unique common fixed point.

Proof: ?? ? is a Cauchy sequence?

Since ? is a continuous mapping from ? to ?(?) ? ?(?) so for ? 1 ? ?, there exists any ? 0 ? ? such that ?? 0 = ?? 1 and ?? 0 = ?? 1

On keep repeating this process for different ? 1 and ? 0 , we get a sequence {? ? } such that In general, we get ?(?? ?+1 , ?? ?+2 , ??) ? ?(?? ? , ?? ?+1 , ?), ? ? > 0 ? (???)

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

Since the space ? is complete, so there exists some ? ? ? such that lim The mappings ? and ? are continuous. ? is continuous from ? to ?(?) ? ?(?).

Clearly, ?(?) ? ?(?) and ?(?) ? ?(?)

This implies that ?(?) ? ?(?) ? ?(?).