# 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][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. # II. PRELIMINARIES 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: ? 1 ? ? 2 â??" ??(? 1 ) ? ??(? 2 ) and ??(? 1 ) ? ??(? 2 ) 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; 0 ? ? 1 ? ? 2 ? |? 1 | < |? 2 |, ? 1 ? ? 2 , ? 2 ? ? 3 ? ? 1 ? ? 3 . Def.2.2.[21]. Complex-Valued Metric Space (CVMS) 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 (1) ??? {? 1 , ? 2 } = ? 2 â??" ? 1 ? ? 2 (2) ? 1 ? ??? {? 2 , ? 3 } ? ? 1 ? ? 2 or ? 1 ? ? 3 And the 'min' functions can be defined as (1) ??? {? 1 , ? 2 } = ? 1 â??" ? 1 ? ? 2 (2) ??? {? 1 , ? 2 } ? ? 3 ? ? 1 ? ? 3 or ? 2 ? ? 3 . 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. # 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 ? 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) # III. MAIN RESULTS 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 ?(?) ? ?(?) ? ?(?). ![CF1) ?(?, ?, ?) > 0, (CF2) ?(?, ?, ?) = ? ?? for all ? > 0 â??" ? = ?, (CF3) ?(?, ?, ?) = ?(?, ?, ?), (CF4) ?(?, ?, ?) * ?(?, ?, ?) ? ?(?, ?, ? + ?), (CF5) ?(?, ?, . ) ? (0, ?) ? ? ? ? ?? is continuous, for all ?, ?, ? ? ?, ?, ? > 0, ? ? ? [0, 1] and ? ? [0, ? 2 ].](image-2.png "(") ![? = ?? ?+1 and ?? ? = ?? ?+1 Or ?? 2? = ?? 2?+1 and ?? 2? = ?? 2?+1 , ? = 1,2,3, ? On setting ? = ? 2? and ? = ? 2?+1 in (3.11), we get for ? = 1,2,3, ? ?(?? 2? , ?? 2?+1 , ??) ? ??? { ?(?? 2?+1 , ?? 2?+1 , ?), ?(?? 2? , ?? 2? , ?), ?(?? 2? , ?? 2?+1 , ?)} ?(?? 2? , ?? 2?+1 , ??) ? ??? { ?(?? 2? , ?? 2?+1 , ?), ?(?? 2?-1 , ?? 2? , ?), ?(?? 2?-1 , ?? 2? , ?)} ?(?? 2? , ?? 2?+1 , ??) ? ??? { ?(?? 2? , ?? 2?+1 , ?), ?(?? 2?-1 , ?? 2? , ?)} ? (?) Now suppose ??? { ?(?? 2? , ?? 2?+1 , ?), ?(?? 2?-1 , ?? 2? , ?)} = ?(?? 2? , ?? 2?+1 , ?) Then by (?), we have ?(?? 2? , ?? 2?+1 , ??) ? ?(?? 2? , ?? 2?+1 , ?) By lemma (4.1) or (5.1), we have ?? 2? = ?? 2?+1 Which is not possible Hence by (?), we must have ?(?? 2? , ?? 2?+1 , ??) ? ?(?? 2?-1 , ?? 2? , ?), ? ? > 0 ? (??)](image-3.png "") 2123323331321![follows that ?? = ?? = ??, and ?(??, ? 2 ?, ??) ? ??? { ?(???, ???, ?), ?(??, ??, ?), ?(??, ???, ?)} ?(??, ? 2 ?, ??) ? ?(??, ???, ?) ?(??, ? 2 ?, ??) ? ?(??, ? 2 ?, ?) ? ?(??, ? 2 ?, ??) ? ? ( ??, ? 2 ?, ? ? ? ) ? (??) On taking ? ? ?, then by lemma (4.1), we have; ?? = ? 2 ? London Journal of Research in Science: Natural and Formal This implies that ?? = ? Thus ? is a common fixed point of ?, S, and ?. Uniqueness: -let ?(? ?) be another fixed point of ?, S, and ?. Then, by (3.11), we have ?(??, ??, ??) ? ??? { ?(??, ??, ?), ?(??, ??, ?), ?(??, ??, ?)} Which implies that ?( ?, ?, ??) ? ??? { ? ?? , ? ?? , ?( ?, ?, ?) } As ?(?, ?, ?) ? ? ? ? ?? , ? ? ? [0, 1] and ? ? [0, ? , also ?(?, ?, ?) ? ? ?? Then certainly we get, ??? { ? ?? , ? ?? , ?(?, ?, ?)} = ?(?, ?, ?) ?(?, ?, ??) ? ?(?, ?, ?) Which implies that ? = ?. As a result, p is unique. Ex. 3.1. Let ? = [3,21] with the metric ? defined by ?(?, ?) = |? -?|, ??, ? ? ?. For all ?, ? ? ? and ? ? (0, ?), we define ?(?, ?, ?) = ? ?? [ ? ?+?(?,?) ] or ?(?, ?, ?) = ? ?? [ ? ??+?(?,?) ] , ? = , and ? -norm ? * ? is defined as ? * ? = ??? {?, ?} where ?, ? ? ? ? ? ?? , for ? ? ? [0, 1] and ? ? [0, ? Here, lim ??? ?(?, ?, ?) = ? ?? , for all ?, ? ? ?. (?, ?, * ) is a CVCFMS with a given ?-norm * . ?, ? ? ? ? ? are defined as: ?(?) = { 3; ?? ? = ? ? 21 , and ?(?) = { 3; ?? ? = ?: ? ? ?(?) ? ?(?) as:](image-4.png "2 ] 1 2 2 ]. 3 ? 3 + 2 ; 3 < 3 2? 3 + 1 ; 3 < ? ? 21 And") Def.2.5. [25]. Complex Valued Continuous t-normA binary operation * ? ? 2 ], iscalled complex valued continuous t-norm if it satisfies the following conditions:(1) * is associative and commutative,(2) * is continuous,(3) ? 2].(iii) ? * ? = {min{?, ?} , ?ð??" max{?, ?} = ? ?? ; 0, ?????????,for a fix ? ? [0,? 2]. ? ? [0, 1]. Ex.2.5. [25]. The following binary operations defined in (i), (ii) and (iii) are complex valued continuous t-norm (i) ? * ? = ??? (?, ?). (ii) ? * ? = ??? (? + ? -? ?? , 0), for a fix ? ? [0, Def.2.6. [25]. Complex Valued Fuzzy Metric Spaces (CVFMS) Lemma 2.7 [25]. Let (?, ?, * ) be a CVFMS such that lim ????(?, ?, ?) = ? ?? , for all ?, ? ??, if ?(?, ?, ??) ? ?(?, ?, ?), for all ?, ? ? ?, 0 < ? < 1, ? ? (0, ?) then ? = ?.Lemma 2.8 [25]. Let {? ? } be a sequence in a CVFMS (?, ?, * ) with lim ????(?, ?, ?) = ? ?? ,for all ?, ? ? ?. If there exists a number ? which lies on (0, 1)such that?(? ?+1 , ? ?+2 , ??) ? ?(? ? , ? ?+1 , ?), ? ? > 0, ? = 0, 1, 2, . .. Then {? ? } is a Cauchysequence 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]. 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