Neutrosophic Quadruple Metric Spaces and Neutrosophic Quadruple Normed Spaces

Neutrosophic Sets and Systems, Vol. 98, 2026 University of New Mexico Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Neutrosophic Quadruple Metric Spaces and Neutrosophic Quadruple Normed Spaces Memet Şahin1*, Arif Sarıoğlan 1 and Amanzholova Alina Bolatkyzy 2 1* Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey. [email protected] 1 Department of Mathematics, Gaziantep University, Gaziantep 27310, Turkey. [email protected] 2 Department of Mathematics, Faculty of Natural Sciences, International Kazakh-Turkish University, 161200, Republic of Kazakhstan. [email protected] * Correspondence: [email protected]; Tel.: +905432182646 Abstract: This study establishes fundamental results in the emerging domains of neutrosophic quadruple metric spaces and neutrosophic quadruple normed spaces. Building upon the recent definition of neutrosophic quadruple metric spaces, the first primary contribution is the development of fixed-point theory within this generalized framework. Specifically, we formulate and rigorously prove an analogue of the Banach contraction principle tailored for neutrosophic quadruple metric spaces. Furthermore, we establish additional fixed-point theorems applicable in this context, extending foundational results from classical metric spaces and simpler neutrosophic structures to handle the increased complexity and uncertainty modeled by quadruple-valued neutrosophic sets. The second major contribution involves the algebraic generalization of normed spaces. We introduce the novel concept of neutrosophic quadruple normed spaces. Within these spaces, we define an appropriate norm structure capable of measuring the "magnitude" of vectors characterized by quadruple-valued neutrosophic components. We systematically investigate and establish various fundamental properties of this newly defined norm, exploring concepts such as statistical convergence, Cauchy statistical convergence, boundedness, and continuity within the neutrosophic quadruple normed spaces. Additionally, we address the specific case of neutrosophic quadruple vector spaces, defining and analyzing the corresponding norm structure in this specialized context. The results generalize and extend prior work in fuzzy, intuitionistic fuzzy, and standard neutrosophic metric and normed spaces. Keywords: fixed points for neutrosophic quadruple metric space; neutrosophic quadruple normed space; statistical convergence to the neutrosophic quadruple norm. 1. Introduction Fixed point theory stands as a cornerstone of nonlinear functional analysis, witnessing remarkable growth in both theoretical depth and practical application. Its significance extends far beyond the boundaries of pure mathematics, permeating diverse fields such as differential equations, optimization theory, game theory, economics, computer science, and engineering. This widespread utility has naturally spurred intense research activity, leading to the development of numerous fixed-point theorems tailored to increasingly sophisticated mathematical structures. Neutrosophic Sets and Systems, Vol. 98, 2026 66 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems The foundation of much of this work lies in the theory of metric spaces. Seminal contributions by T. Zamfirescu [1] established crucial fixed-point results within this classical setting. As mathematical frameworks evolved to model greater complexity and uncertainty, researchers explored generalizations of metric spaces. This led to the investigation of fixed-point theorems in fuzzy metric (shortly FM) spaces [2], capturing vagueness through membership degrees, and subsequently in intuitionistic FM spaces [3], which incorporate both membership and non-membership. More recently, the advent of neutrosophic metric spaces [8], built upon the neutrosophic logic principle of characterizing truth, falsehood, and indeterminacy simultaneously, provided a richer framework for handling incomplete, inconsistent, and uncertain information, yielding novel fixed-point results. The ongoing quest for structures capable of representing higher-dimensional uncertainty culminated in the definition of neutrosophic quadruple metric (shortly NQM) spaces [6]. This innovative framework, extending the representational capacity of neutrosophic sets to quadruples, immediately presented the compelling challenge of establishing fixed point theorems within this new context. Addressing this gap forms a primary objective of the present study. Consequently, the first part of this paper is dedicated to exploring contraction principles and fixed-point theorems specifically within NQM spaces. We establish and prove an analogue of the fundamental Banach contraction principle in this setting and investigate other relevant fixed-point results. Parallel to the development of metric structures in uncertain environments, the concept of normed vector spaces has also been generalized. The norm, fundamentally quantifying the "length" or magnitude of a vector, is essential for analysis in vector spaces. Significant efforts have been made to define appropriate norms for vectors residing in fuzzy [11], intuitionistic fuzzy [12], and neutrosophic vector spaces, enabling the study of functional analysis in these generalized settings. In studies [14] and [15], previous studies have examined statistical convergence in standard normed spaces and intuitionistic fuzzy normed spaces. In this study, we generalize this concept to neutrosophic quadruple normed spaces. [16] Deli et al. defined n-valued neutrosophic trapezoidal numbers with similarity measures and its properties. [17] Sahin et al. investigated the extension principles of neutrosophic multi-sets and cut sets and algebraic operators. [18] Ulucay defined similarity function of trapezoidal fuzzy multi-numbers. [19] Bakbak and Ulucay analyzed Q-neutrosophic soft expert multiset and its set operations (like union, intersection,complement,subset). [20] Baser and Ulucay studied an application of neutrosophic soft sets and its properties. [21] Baser and Ulucay investigated effective Q-neutrosophic soft expert sets on an application. No prior studies address properties such as metric, continuity, and completeness in NQS. Various applications of neutrosophic sets are discussed in this essay [22-37]. Building upon this progression, the second part of our work introduces the concept of neutrosophic quadruple normed (shortly NQN) spaces. We define a suitable norm structure for vector spaces where the vectors themselves possess neutrosophic quadruple characteristics. Furthermore, we extend this concept specifically to neutrosophic quadruple vector spaces, defining and examining the properties of the norm within this specialized context. This study thus contributes to two expanding frontiers: establishing fundamental fixed-point results in the novel setting of NQM spaces and initiating the development of norm theory for neutrosophic quadruple sets. 2. Preliminaries Definition 2.1 [4] Let be binary operation. Then is called a t-norm (or TN) if it satisfies the following properties for all elements I. Identity: Neutrosophic Sets and Systems, Vol. 98, 2026 67 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems II. Monotonicity: If and , then . III. Symmetry: IV. Associativity: V. Continuity: The operation is continous. Definition 2.2 [5] Let be binary operation. Then is called a t-conorm (or TC) if it satisfies the following proporties for all elements : I. Identity: II. Monotonicity: . III. Symmetry: . IV. Associativity: . V. Continuity: The operation is continous. Definition 2.3 [6] , and let T represent a truth-membership degree, I an indeterminacy-membership degree, and F a falsity-membership degree. A neutrosophic quadruple set element D is defined as: Here, is called the known segment of , while the expression constitutes its unknown segment. Alternatively, the quadruple number can be represented as an ordered tuple: is neutrosophic quadruple set. Definition 2.4 [7] Let be a non-empty set, , and be a neutrosophic metric on where represent truth, indeterminacy, and falsity degrees respectively. Let be a metric, be positive constants, and denote a t-norm (TN) and t-conorm (TC) respectively. The neutrosophic quadruple metric (NQM) is defined as: Neutrosophic Sets and Systems, Vol. 98, 2026 68 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Definition 2.5 [8] Let be a NM on . The mapping is defined neutrosophic contraction (NC) if there exists such that for each and . Definition 2.6 [8] Let be a NM on and let be a NC mapping. Then there exists such that . That is, is called neutrosophic fixed point (NFP) of . Theorem 2.7 [10] Let K denote one of the following fields: , , or (where p is a prime). Define the neutrosophic quadruple group as the set equipped with addition (+). Then forms a neutrosophic quadruple vector space over K, where the operation ∙∙ (scalar multiplication) is a specially defined bilinear map from to NQ. Definition 2.8 [13] Let Q be a vector space over a field K (typically or ). Let Be a mapping, denoted for each and by , where , , represent the degrees of truth, indeterminacy, and falsity of the assertion "the norm of p is less than or equal to t", respectively. Let be a t-norm and ▶ be a t-conorm. The 4-tuple is called a neutrosophic normed space if for all and all positive real numbers the following conditions hold: I. II. Neutrosophic Sets and Systems, Vol. 98, 2026 69 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems III. if and only if IV. V. VI. is continuous non-decreasing function, VII. VIII. if and only if IX. X. XI. is continuous non-decreasing function, XII. XIII. if and only if XIV. XV. XVI. is continuous non-decreasing function, XVII. XVIII. If , then and Then is called neutrosophic norm (shortly NM). Definition 2.9 [14] Let be a subset of the natural numbers . The asymptotic density of , denoted by , is defined as where denotes the cardinality of the set. Neutrosophic Sets and Systems, Vol. 98, 2026 70 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Let be a set and let be number pair of in such that and . Then the two dimensional asymptotic density can be define as . A sequence is statistically convergent to if for every , the asymptotic density of the set is zero. That is, A double sequence is statistically convergent to if for every , the double asymptotic density of the set is zero. That is, Definition 2.10 [14] A sequence is called a statistically Cauchy sequence if for every , Definition 2.11 [15] Let be an intuitionistic fuzzy normed space. A double sequence is statistically convergent to with respect to the intuitionistic fuzzy norm if for every and every , and . That is Definition 2.12 [15] Let be an intuitionistic fuzzy normed space. A double sequence is statistically Cauchy with respect to the intuitionistic fuzzy norm if for every and every , and . Neutrosophic Sets and Systems, Vol. 98, 2026 71 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems That is 3. Fixed-Point Theorems In Neutrosophic Quadruple Metric Spaces Definition 3.1 Let be a NQM space. The mapping is called neutrosophic quadruple contraction if there exist such that for each and . We will define neutrosophic quadruple banach contraction theorem. Theorem 3.1 Let be a neutrosophic quadruple metric space (NQM). If is a neutrosophic quadruple contraction mapping, then there exists an element such that . This element c is called a neutrosophic quadruple fixed point of T. Proof Let and . By a induction we get for each and . For any ,we get Neutrosophic Sets and Systems, Vol. 98, 2026 72 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems If we apply the limit conditions for the functions in the neutrosophic metric space definition, we get and That is, cauchy sequence. Hence is convergent. We define a limit point for sequence. We get Neutrosophic Sets and Systems, Vol. 98, 2026 73 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Thus we get a fixed point. Now to get uniqueness, we assume for any . Then t when So . Corollary 3.1 Let be a complete NQM space and let be a neutrosophic quadruple contraction mapping. Then T possesses a unique fixed point in . Now we will show edelstein contraction theorem for NQM spaces. Theorem 3.2 Let be a NQM space and let be a neutrosophic quadruple contraction mapping. if for all Neutrosophic Sets and Systems, Vol. 98, 2026 80 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Corollary 4.1 In a NQN space, every compact set is closed and neutrosophic quadruple bounded. The proof of this corollary can be easily obtained from the above theorems and definitions. Definition 4.5 Let be a neutrosophic quadruple normed space. Then a sequence is called to statistically convergent to with to nutrosophic quadruple norm hold on that for all and , and . That is Example 4.3 Let be classic norm for real numbers and for the t-norm is and t-conorm is for every For and every is defined as Then is a neutrosophic quadruple normed space. For the sequence is defined by Then, we get Neutrosophic Sets and Systems, Vol. 98, 2026 81 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems Definition 4.6 Let be a neutrosophic quadruple normed space. Then a double sequence is called to statistically cauchy convergent to with to neutrosophic quadruple norm if for all and , there exist and such that for every and . Definition 4.6 Let be a neutrosophic quadruple normed space. A double sequence is statistically Cauchy with respect to the neutrosophic quadruple norm if for every and and . That is 5. Conclusions In this study, we have defined the structure of neutrosophic quadruple normed spaces based on the neutrosophic quadruple set. Within this framework, we established the neutrosophic quadruple Banach contraction theorem and proved fundamental fixed-point theorems in neutrosophic quadruple metric spaces. We obtained definitions of statistical convergence and Cauchy statistical convergence for neutrosophic quadruple normed spaces. This research successfully bridges a critical theoretical gap. It not only provides foundational fixed-point results for the newly introduced neutrosophic quadruple metric spaces but also pioneers the development of the theory of neutrosophic quadruple normed spaces. These contributions furnish essential mathematical tools and open significant avenues for advanced analysis under Neutrosophic Sets and Systems, Vol. 98, 2026 82 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems complex, multifaceted uncertainty. Collectively, the results presented offer a natural and substantial extension of existing knowledge in fixed point theory and functional analysis within fuzzy, intuitionistic fuzzy, and neutrosophic settings. Abbreviations FS Fuzzy set FN Fuzzy norm NN Neutrosophic Norm NQM Neutrosophic quadruple metric NQN Neutrosophic quadruple norm TN Continuous t-norm TC Continuous t-conorm Funding: No funding or institutional support was received during the preparation of this article. Acknowledgments: The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper. Conflicts of Interest: The authors declare no conflict of interest References 1. Zamfirescu, Tudor. "Fix point theorems in metric spaces." Archiv der Mathematik 23 (1972): 292-298. 2. Gregori, Valentın, and Almanzor Sapena. "On fixed-point theorems in fuzzy metric spaces." Fuzzy sets and systems 125.2 (2002): 245-252. 3. Alaca, Cihangir, Duran Turkoglu, and Cemil Yildiz. "Fixed points in intuitionistic fuzzy metric spaces." Chaos, Solitons & Fractals 29.5 (2006): 1073-1078. 4. Dubois, Didier, and Henri Prade. "A review of fuzzy set aggregation connectives." Information sciences 36.1-2 (1985): 85-121. 5. Dubois, Didier J. Fuzzy sets and systems: theory and applications. Vol. 144. Academic press, 1980. 6. Smarandache, F. Neutrosophic quadruple numbers, refined neutrosophic quadruple numbers, absorbance law, and the multiplication of neutrosophic quadruple numbers. Neutrosophic Sets Syst. 2015, 10, 1. 7. Şahin, Memet, and Arif Sarıoğlan. "Neutrosophic Quadruple Metric Spaces." Symmetry 17.7 (2025): 1096. 8. Kirişci, Murat, Necip Şimşek, and Mahmut Akyiğit. "Fixed point results for a new metric space." arXiv preprint arXiv:1910.03573 (2019).7 9. Akinleye¹, S. A., F. Smarandache, and A. A. A. Agboola. "On neutrosophic quadruple algebraic structures." Neutrosophic Sets and Systems, vol. 12/2016: A Quarterly International Journal in Information Science and Engineering (2016): 122. Neutrosophic Sets and Systems, Vol. 98, 2026 83 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems 10. WB, Vasantha Kandasamy, Ilanthenral Kandasamy, and Florentin Smarandache. "Neutrosophic quadruple vector spaces and their properties." Mathematics 7.8 (2019): 758. 11. Xiao, J. Z., & Zhu, X. H. (2003). Fuzzy normed space of operators and its completeness. Fuzzy sets and Systems, 133(3), 389-399. 12. Saadati, R., & Park, J. H. (2006). On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals, 27(2), 331-344. 13. Kirişçi, Murat, and Necip Şimşek. "Neutrosophic normed spaces and statistical convergence." The Journal of Analysis 28.4 (2020): 1059-1073. 14. Fridy, John A. "On statistical convergence." Analysis 5.4 (1985): 301-314. 15. Mursaleen, M., & Mohiuddine, S. A. (2009). Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos, Solitons & Fractals, 41(5), 2414-2421. 16. Deli, İ., Uluçay, V., & Polat, Y. (2022). N-valued neutrosophic trapezoidal numbers with similarity measures and application to multi-criteria decision-making problems. Journal of Ambient Intelligence and Humanized Computing, 1-26. https://doi.org/10.1007/s12652-021-03294-7 17. Sahin, M., Deli, I., & Uluçay, V. (2017). Extension principle based on neutrosophic multi-fuzzy sets and algebraic operations. Infinite Study 18. Uluçay, V. (2020). A new similarity function of trapezoidal fuzzy multi-numbers based on multi-criteria decision making. J Inst Sci Technol, 10(2), 1233-1246. https://doi.org/10.21597/jist.644794 19. Bakbak, D., & Uluçay, V. (2019). Chapter Eight Multiple Criteria Decision Making in Architecture 20. Based on Q-Neutrosophic Soft Expert Multiset. Neutrosophic Triplet Structures, 90. 20.Başer, Z., & Ulucay, V. (2025). Energy of a neutrosophic soft set and its applications to multi-criteria decision-making problems. Neutrosophic Sets and Systems, 79(1), 28. 21. Başer, Z., & Uluçay, V. (2024). Effective Q-Neutrosophic Soft Expert Sets and Its Application in Decision Making. Algebraic Structures In the Universe of Neutrosophic: Analysis with Innovative Algorithmic Approaches, 147. 22. Deli, I., Ulucay, V., & Başer, Z. (2025). Neutrosophic Inference Systems Using Takagi-Sugeno-Kang Model and Its Application. Neutrosophic Sets and Systems, 88, 1019-1036. 23. Ulucay, V., Kılıç, A., Sahin, M., & Deniz, H. (2019). A new hybrid distance-based similarity measure for refined neutrosophic sets and its application in medical diagnosis. Infinite Study. 24. Ulucay, V., Sahin, M., & Olgun, N. (2018). Time-neutrosophic soft expert sets and its decision making problem. Infinite Study. 25. Uluçay, V., & Şahin, M. (2020). Decision-making method based on neutrosophic soft expert graphs. In Neutrosophic Graph Theory and Algorithms (pp. 33-76). IGI Global Scientific Publishing. 26. Şahin, M., Ulucay, V., & Menekşe, M. (2018). Some New Operations of (α, β, γ) Interval Cut Set of Interval Valued Neutrosophic Sets. Journal of Mathematical & Fundamental Sciences, 50(2). 27. Uluçay, V., & Şahin, M. (2024). Intuitionistic fuzzy soft expert graphs with application. Uncertainty Discourse and Applications, 1(1), 1-10. 28. Uluçay, V., & Şahin, M. (2019). Neutrosophic Multigroups and Applications. Mathematics, 7(1), 95. 29. Şahin, M & Uluçay, V. (2019). Fuzzy Soft Expert Graphs With Application. Asian Journal of Mathematics and Computer Research, 26 (4), pp.216-229. Neutrosophic Sets and Systems, Vol. 98, 2026 84 Şahin et al., Neutrosophic Quadruple Normed Spaces And Fixed-Point Theorems 30. Şahin, M. (2022). Neutro-Sigma Algebras and Anti-Sigma Algebras. Neutrosophic Sets and Systems, 51, 908-922. 31. Şahin, M., Ulucay, V., Olgun, N. (2017). Normed Z-Modules. International Journal of Pure and Applied Mathematics, 112(2), 425-435. 32. Uluçay, V., Deli, I., & Sahin, M. (2019). Intuitionistic trapezoidal fuzzy multi-numbers and its application to multi-criteria decision-making problems, Complex Intell. Syst., 5, 65–78. 33. Sahin, M. (2021). Neutrosophic Multigroup Homomorphism and Some of its Properties. International Journal of Neutrosophic Science, 17(2), 110 - 126. 34. Uluçay, V., Kocasakal, E. G., & Başer, Z. (2025). A hybrid decision-making technique of CoCoSo and entropy methods on trapezoidal fuzzy multi numbers and its application. Journal of Fuzzy Extension and Applications, e227153. 35. Uluçay, V., Deli, İ., & Başer, Z. (2025). Intuitionistic fuzzy Takagi–Sugeno–Kang non-linear inference system and its applications to greenhouse control system in agriculture. International Journal of Information Technology, 1-10. 36. Uluçay, V., Başer, Z., & Edalatpanah, S. A. (2025). Euclidean distance on treesoft sets and application for electric vehicle selection. Journal of Fuzzy Extension and Applications, e232762. 37. Şahin, M & Uluçay, V. (2020). Soft Maksimal Ideals on Soft Normed Rings. In Quadruple Neutrosophic Theory and Applications, 203 – 212, Pons Editions. Received: July 6, 2025. Accepted: Dec 16, 2025