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}{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \affil[1]{ } \renewcommand\Authands{ and } \date{\small \em Received: 1 January 1970 Accepted: 1 January 1970 Published: 1 January 1970} \maketitle \begin{abstract} This work demonstrates certain standard fixed point theorems on complex-valued fuzzy metric spaces. We show certain fixed point findings in the situation of complex-valued fuzzy metric spaces, inspired by Singh et al. [25].To begin, we extend some well-known existing conclusions from metric spaces to complex-valued fuzzy metric spaces and then prove them in the complex-valued complete fuzzy metric space context. We provide an example that supports our main result and supports our hypotheses. \end{abstract} \keywords{} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \begin{textblock*}{10cm}(1.05cm,3cm) {{\textit{CrossRef DOI of original article:}} \underline{}} \end{textblock*}\let\tabcellsep& \section[{I. INTRODUCTION}]{I. INTRODUCTION}\par In 1965, Zadeh \hyperref[b3]{[3]} coined the term "fuzzy set." Following that, a slew of authors worked on fuzzy sets, expanding the fuzzy set theory and its applications \hyperref[b4]{[4]}\hyperref[b5]{[5]}\hyperref[b6]{[6]}. The idea of fuzzy metric spaces was given by Kramosil and Michalik \hyperref[b7]{[7]}. After then, George and Veeramani \hyperref[b9]{[9]} updated this idea. Grabiec \hyperref[b8]{[8]} investigated fuzzy metric space fixed-point theory. The idea of complex-valued metric spaces was introduced by Azam et al. \hyperref[b21]{[21]}.\par Verma et al. \hyperref[b23]{[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. {\ref 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.\par 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. \section[{II. PRELIMINARIES}]{II. PRELIMINARIES}\par Def.2.1. \hyperref[b21]{[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 )\par Hence ? 1 ? ? 2 if one of the following satisfies;\par 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 )\par 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.\par 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)\par Let ? be a non-empty set. Assume that the mappings ?: ? × ? ? ? satisfies: (CV1) 0 ? ?(?, ?), for all ?, ? ? ? and ?(?, ?) = 0 iff ? = ? ;\par (CV2) ?(?, ?) = ?(?, ?), for all ?, ? ? ? ;\par (CV3) ?(?, ?) ? ?(?, ?) + ?(?, ?), for all ?, ?, ? ? ? Then ? is called a complex-valued metric on ?, and (?, ?) is called a CVMS. Def.2.3. \hyperref[b23]{[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\par And the 'min' functions can be defined as(1) ??? \{? 1 , ? 2 \} = ? 1 â??" ? 1 ? ? 2 (2) ??? \{? 1 , ? 2 \} ? ? 3 ? ? 1 ? ? 3 or ? 2 ? ? 3 .\par Following Zadeh's \hyperref[b3]{[3]} contribution to fuzzy set theory, a number of scholars \hyperref[b4]{[4]}\hyperref[b5]{[5]}\hyperref[b6]{[6]} contributed to the field's basics and core theories.\par Buckley \hyperref[b10]{[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. \hyperref[b1]{[1]} expanded fuzzy sets to complex fuzzy sets in this chain. \section[{Singh et al. [25], inspired by Ramot et al. [1,}]{Singh et al. [25], inspired by Ramot et al. [1,}\par ] 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.\par We establish certain fixed-point conclusions in the situation of complex -valued fuzzy metric spaces, inspired by {\ref 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. \hyperref[b1]{[1]}. The complex fuzzy set ? is given by ? = \{(?, ? ? (?)) ? ? ? ?\}.\par 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.\par Fisher \hyperref[b24]{[24]} established the following theorem in complete metric space for three mappings.\par Theorem A \hyperref[b24]{[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;\par ?(??, ??) ? ? ?(??, ??) for all ?, ? ? ? and 0 < ? < 1. Indeed ?, ? and ? have a unique common fixed point.\par We can now extend the preceding theorem/result to complex-valued complete fuzzy metric space as follows:\par 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.\par ?(??, ??, ??) ? ???\{ ?(??, ??, ?), ?(??, ??, ?), ?(??, ??, ?)\} for all ?, ? ? ?, ? ? (0, ?) and 0 < ? < 1 ? (3.11) \section[{III. MAIN RESULTS}]{III. MAIN RESULTS}\par Additionally, lim ??? ?(?, ?, ?) = ? ?? , for all ?, ? ? ? and ? ? [0, Then ?, ?, and ? have a unique common fixed point.\par Proof: ?? ? is a Cauchy sequence?\par Since ? is a continuous mapping from ? to ?(?) ? ?(?) so for ? 1 ? ?, there exists any ? 0 ? ? such that ?? 0 = ?? 1 and ?? 0 = ?? 1\par On keep repeating this process for different ? 1 and ? 0 , we get a sequence \{? ? \} such that In general, we get ?(?? ?+1 , ?? ?+2 , ??) ? ?(?? ? , ?? ?+1 , ?), ? ? > 0 ? (???)??\par Hence by lemma (4.2), \{?? ? \} is a Cauchy sequence in ?.\par Since the space ? is complete, so there exists some ? ? ? such that lim The mappings ? and ? are continuous. ? is continuous from ? to ?(?) ? ?(?).\par Clearly, ?(?) ? ?(?) and ?(?) ? ?(?)\par This implies that ?(?) ? ?(?) ? ?(?).\begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-2.png} \caption{\label{fig_0}(}\end{figure} \begin{figure}[htbp] \noindent\textbf{}\includegraphics[]{image-3.png} \caption{\label{fig_2}}\end{figure} \begin{figure}[htbp] \noindent\textbf{2123323331321}\includegraphics[]{image-4.png} \caption{\label{fig_3}2 ] 1 2 2 ]. 3 ? 3 + 2 ; 3 < 3 2? 3 + 1 ; 3 < ? ? 21 And}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.6440184049079754\textwidth}P{0.14340490797546013\textwidth}P{0.04432515337423312\textwidth}P{0.005214723926380368\textwidth}P{0.007822085889570552\textwidth}P{0.005214723926380368\textwidth}} \multicolumn{2}{l}{Def.2.5. [25]. Complex Valued Continuous t-norm}\tabcellsep \tabcellsep \tabcellsep \\ \multicolumn{6}{l}{A binary operation * ? ? 2 ], is}\\ \multicolumn{6}{l}{called complex valued continuous t-norm if it satisfies the following conditions:}\\ \multicolumn{2}{l}{(1) * is associative and commutative,}\tabcellsep \tabcellsep \tabcellsep \\ \multicolumn{2}{l}{(2) * is continuous,}\tabcellsep \tabcellsep \tabcellsep \\ \multicolumn{3}{l}{(3) ? 2}\tabcellsep ].\tabcellsep \\ (iii) ? * ? = \{\tabcellsep min\{?, ?\} , ?ð??" max\{?, ?\} = ? ?? ; 0, ?????????,\tabcellsep \multicolumn{2}{l}{for a fix ? ? [0,}\tabcellsep ? 2\tabcellsep ].\end{longtable} \par \begin{quote} ? ? [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)\end{quote} \caption{\label{tab_0}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.7512061403508772\textwidth}P{0.06337719298245613\textwidth}P{0.035416666666666666\textwidth}} Lemma 2.7 [25]. Let (?, ?, * ) be a CVFMS such that lim ???\tabcellsep \multicolumn{2}{l}{?(?, ?, ?) = ? ?? , for all ?, ? ?}\\ \multicolumn{3}{l}{?, if ?(?, ?, ??) ? ?(?, ?, ?), for all ?, ? ? ?, 0 < ? < 1, ? ? (0, ?) then ? = ?.}\\ \multicolumn{2}{l}{Lemma 2.8 [25]. Let \{? ? \} be a sequence in a CVFMS (?, ?, * ) with lim ???}\tabcellsep ?(?, ?, ?) = ? ?? ,\\ \multicolumn{3}{l}{for all ?, ? ? ?. If there exists a number ? which lies on (0, 1)such that}\\ \multicolumn{3}{l}{?(? ?+1 , ? ?+2 , ??) ? ?(? ? , ? ?+1 , ?), ? ? > 0, ? = 0, 1, 2, . .. Then \{? ? \} is a Cauchy}\\ sequence in ?.\tabcellsep \end{longtable} \par \begin{quote} \par \par The following theorem was established bySingh et al. [25], which is the resetting of the Banach contraction principle in CVFMS. Theorem 2.7 [25]. Let (?, ?, * ) be a CVFMS such that lim ??? ?(?, ?, ?) = ? ?? , ? ?, ? ? ?,\end{quote} \caption{\label{tab_1}}\end{figure} \label{foot_0}\footnote{\label{foot_0} Volume 23 | Issue 2 | Compilation 1.0 © 2023 Great Britain Journal Press} \backmatter \begin{bibitemlist}{1} \bibitem[Ramot et al. ()]{b1}\label{b1} \textit{}, D Ramot , R Milo , M Friedman , A Kandel . \textit{IEEE Transactions of Fuzzy Systems} 2002. 10 (2) p. . \bibitem[Singh et al. ()]{b24}\label{b24} ‘A novel framework of complex-valued fuzzy metric spaces and fixed-point theorems’. D Singh , V Joshi , M Imdad , P Kumam . \textit{Journal of Intelligent and fuzzy system} 2016. 30 p. . \bibitem[Azam et al. ()]{b20}\label{b20} ‘Common fixed point theorems in complex valued metric spaces’. A Azam , B Fisher , M Khan . \textit{Numerical Functional Analysis and Optimization} 2011. 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