A generalized framework for $ \varsigma $-neutrosophic fuzzy metric spaces and related fixed-point theorems

Xiu-Liang Qiu; Ömer Kişi; Mehmet Gürdal; Qing-Bo Cai; Xiu-Liang Qiu; Ömer Kişi; Mehmet Gürdal; Qing-Bo Cai

doi:10.3934/math.20251247

AIMS Mathematics

2025, Volume 10, Issue 12: 28347-28373. doi: 10.3934/math.20251247

Previous Article Next Article

Research article

A generalized framework for $ \varsigma $-neutrosophic fuzzy metric spaces and related fixed-point theorems

1.
Chengyi College, Jimei University, Xiamen 361021, China
2.
Department of Mathematics, Bartin University, Bartın, Turkey
3.
Department of Mathematics, Süleyman Demirel University, 32260, Isparta, Turkey
4.
School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China

Received: 09 October 2025 Revised: 19 November 2025 Accepted: 26 November 2025 Published: 03 December 2025
MSC : 47H10, 54A40, 54E70

This paper introduces $ \varsigma $-neutrosophic fuzzy metric spaces ($ \varsigma $-NFMSs), a significant generalization of neutrosophic fuzzy metric spaces (NFMSs). By extending the parameter space from a single dimension $ (0, \infty) $ to a multi-dimensional vector space $ (0, \infty)^{\varsigma} $, this framework offers enhanced flexibility for modeling complex systems where uncertainty depends on multiple factors simultaneously. The study investigates the topological properties of $ \varsigma $-NFMSs, rigorously proving that their topology is first-countable and that the associated space is Hausdorff. Furthermore, a generalized fixed-point theorem is established within this new framework, extending previous results in NFMSs.
- neutrosophic fuzzy metric space,
- $ \varsigma $-NFMS,
- Hausdorff topology,
- fixed-point theorem,
- t-norm,
- t-conorm,
- multi-parameter generalization
Citation: Xiu-Liang Qiu, Ömer Kişi, Mehmet Gürdal, Qing-Bo Cai. A generalized framework for $ \varsigma $-neutrosophic fuzzy metric spaces and related fixed-point theorems[J]. AIMS Mathematics, 2025, 10(12): 28347-28373. doi: 10.3934/math.20251247

Related Papers:

Abstract

This paper introduces $ \varsigma $-neutrosophic fuzzy metric spaces ($ \varsigma $-NFMSs), a significant generalization of neutrosophic fuzzy metric spaces (NFMSs). By extending the parameter space from a single dimension $ (0, \infty) $ to a multi-dimensional vector space $ (0, \infty)^{\varsigma} $, this framework offers enhanced flexibility for modeling complex systems where uncertainty depends on multiple factors simultaneously. The study investigates the topological properties of $ \varsigma $-NFMSs, rigorously proving that their topology is first-countable and that the associated space is Hausdorff. Furthermore, a generalized fixed-point theorem is established within this new framework, extending previous results in NFMSs.

References

[1]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
[2]	F. Smarandache, A unifying field in logics: neutrosophic logic. Neutrosophy, neutrosophic set, neutrosophic probability, Rehoboth: American Research Press, 1999.
[3]	K. T. Atanassov, Intuitionistic fuzzy sets, Int. J. Bioautom., 20 (2016), S1–S6.
[4]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3
[5]	R. Biswas, On fuzzy sets and intuitionistic fuzzy sets, NIFS, 3 (1997), 3–11.
[6]	I. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika, 11 (1975), 336–344.
[7]	A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395–399. https://doi.org/10.1016/0165-0114(94)90162-7 doi: 10.1016/0165-0114(94)90162-7
[8]	D. Çoker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst., 88 (1997), 81–89. https://doi.org/10.1016/S0165-0114(96)00076-0 doi: 10.1016/S0165-0114(96)00076-0
[9]	J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fract., 22 (2004), 1039–1046. https://doi.org/10.1016/j.chaos.2004.02.051 doi: 10.1016/j.chaos.2004.02.051
[10]	R. Vinoth, M. Jayalakshmi, Generalization and properties of $\kappa$-intuitionistic fuzzy metric spaces with applications to fixed-point theorems, Contemp. Math., 6 (2025), 903–922. https://doi.org/10.37256/cm.6120255998 doi: 10.37256/cm.6120255998
[11]	O. Kaleva, S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst., 12 (1984), 215–229. https://doi.org/10.1016/0165-0114(84)90069-1
[12]	F. Smarandache, Neutrosophic set–a generalization of the intuitionistic fuzzy set, In: Proceedings of the IEEE International Conference on Granular Computing, 2006, 38–42.
[13]	A. A. Salama, S. A. Alblowi, Neutrosophic set and neutrosophic topological spaces, IOSR J. Math., 3 (2012), 31–35.
[14]	P. A. Ejegwa, S. O. Akowe, P. M. Otene, J. M. Ikyule, An overview on intuitionistic fuzzy sets, Int. J. Sci. Technol. Res., 3 (2014), 142–145.
[15]	P. Majumdar, Neutrosophic sets and its applications to decision making, In: Computational intelligence for big data analysis, Cham: Springer, 2015, 97–115. https://doi.org/10.1007/978-3-319-16598-1_4
[16]	A. Şahiner, M. Gürdal, T. Yigit, Ideal convergence characterization of the completion of linear $n$-normed spaces, Comput. Math. Appl., 61 (2011), 683–689. https://doi.org/10.1016/j.camwa.2010.12.015 doi: 10.1016/j.camwa.2010.12.015
[17]	E. Savaş, Ö. Kişi, M. Gürdal, On statistical convergence in credibility space, Numer. Funct. Anal. Optim., 43 (2022), 987–1008.
[18]	M. Karaev, M. Gürdal, S. Saltan, Some applications of Banach algebra techniques, Math. Nachr., 284 (2011), 1678–1689. https://doi.org/10.1002/mana.200910129 doi: 10.1002/mana.200910129
[19]	M. Gürdal, Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra, Expo. Math., 27 (2009), 153–160. https://doi.org/10.1016/j.exmath.2008.10.006 doi: 10.1016/j.exmath.2008.10.006
[20]	M. T. Karaev, M. Gürdal, M. B. Huban, Reproducing kernels, Englis algebras and some applications, Stud. Math., 232 (2016), 113–141. https://doi.org/10.4064/sm8385-4-2016 doi: 10.4064/sm8385-4-2016
[21]	K. Menger, Statistical metrics, In: Selecta Mathematica II, Vienna: Springer, 2003,433–435. https://doi.org/10.1007/978-3-7091-6045-9_35
[22]	S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl., 83 (1981), 566–569. https://doi.org/10.1016/0022-247X(81)90141-4 doi: 10.1016/0022-247X(81)90141-4
[23]	R. K. Bose, D. Sahani, Fuzzy mappings and fixed point theorems, Fuzzy Sets Syst., 21 (1987), 53–58. https://doi.org/10.1016/0165-0114(87)90152-7 doi: 10.1016/0165-0114(87)90152-7
[24]	C. Alaca, D. Türkoglu, C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos Solitons Fract., 29 (2006), 1073–1078. https://doi.org/10.1016/j.chaos.2005.08.066 doi: 10.1016/j.chaos.2005.08.066
[25]	M. Kirişci, N. Şimşek, Neutrosophic metric spaces, Math. Sci., 14 (2020), 241–248. https://doi.org/10.1007/s40096-020-00335-8
[26]	N. Şimşek, M. Kirişci, Fixed point theorems in neutrosophic metric spaces, Infinite Study, 2019.
[27]	M. Kirişci, N. Şimşek, M. Akyiğit, Fixed point results for a new metric space, Math. Methods Appl. Sci., 44 (2021), 7416–7422. https://doi.org/10.1002/mma.6189 doi: 10.1002/mma.6189
[28]	S. Sowndrarajan, M. Jeyaraman, F. Smarandache, Fixed point results for contraction theorems in neutrosophic metric spaces, Neutrosophic Sets Syst., 36 (2020), 308–318.
[29]	M. Pandiselvi, M. Jeyaraman, New insights into fixed points results in neutrosophic metric like-spaces and their applications, Stat. Optim. Inf. Comput., 2025.
[30]	M. Akram, U. Ishtiaq, K. Ahmad, T. A. Lazăr, V. L. Lazăr, L. Guran, Some generalized neutrosophic metric spaces and fixed point results with applications, Symmetry, 16 (2024), 1–33. https://doi.org/10.3390/sym16080965 doi: 10.3390/sym16080965
[31]	S. Ghosh, R. Bhardwaj, S. Narayan, S. Sonam, Fixed point theorems in neutrosophic fuzzy metric space and a characterization to its completeness, Neutrosophic Sets Syst., 81 (2025), 575–602.
[32]	M. Younis, A. A. N. Abdou, Novel fuzzy contractions and applications to engineering science, Fractal Fract., 8 (2024), 1–19. https://doi.org/10.3390/fractalfract8010028 doi: 10.3390/fractalfract8010028
[33]	S. Jabeena, M. E. Köksal, M. Younis, Convergence results based on graph-Reich contraction in fuzzy metric spaces with application, Nonlinear Anal. Model. Control, 29 (2023), 71–95. https://doi.org/10.15388/namc.2024.29.33668 doi: 10.15388/namc.2024.29.33668
[34]	S. Das, B. K. Roy, M. B. Kar, S. Kar, D. Pamucar, Neutrosophic fuzzy set and its application in decision making, J. Ambient Intell. Humaniz. Comput., 11 (2020), 5017–5029. https://doi.org/10.1007/s12652-020-01808-3 doi: 10.1007/s12652-020-01808-3
[35]	S. Ghosh, Sonam, R. Bhardwaj, S. Narayan, On neutrosophic fuzzy metric space and its topological properties, Symmetry, 16 (2024), 1–14. https://doi.org/10.3390/sym16050613 doi: 10.3390/sym16050613
[36]	S. K. De, R. Biswas, A. R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets Syst., 117 (2001), 209–213. https://doi.org/10.1016/S0165-0114(98)00235-8 doi: 10.1016/S0165-0114(98)00235-8
[37]	M. Younis, M. Öztürk, Some novel proximal point results and applications, Univers. J. Math. Appl., 8 (2025), 8–20.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)