A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- 1.
- ⊗ adheres to both associativity and commutativity properties.
- 2.
- ⊗ needs to be continuous.
- 3.
- For every , it holds that .
- 4.
- For any , if and , then .
- 1.
- ⊕ adheres to both associativity and commutativity properties.
- 2.
- ⊕ needs to be continuous.
- 3.
- For every , it holds that .
- 4.
- For any , if and , then .
- (i)
- for all with ,
- (ii)
- ,
- (iii)
- for all with ,
- (iv)
- , then ,
- (v)
- , where is a permutation function,
- (vi)
- for all ,
- (vii)
- is continuous.
- (viii)
- ,
- (ix)
- for all with ,
- (x)
- , then ,
- (xi)
- , where is a permutation function,
- (xii)
- for all ,
- (xiii)
- is continuous.
- (xiv)
- ,
- (xv)
- for all with ,
- (xvi)
- , then ,
- (xvii)
- , where is a permutation function,
- (xviii)
- for all ,
- (xix)
- is continuous.
- (i)
- converges to ι if and only if , and ; i.e., for all and all , there exists such that and for all (in such a case, we will write .
- (ii)
- A sequence is considered a Cauchy sequence if and only if for every and , there exists an index such thatAdditionally, is a generalized Cauchy sequence if and only if for every and , there exists such thatIn other words,
- (iii)
- GNMS is considered complete if every Cauchy sequence in the space converges to a limit in that space.
3. Fixed-Point Results Using Various -Contractive Mappings
4. () Contractions on Non-Archimedean GNMS
- (1)
- if and only if ;
- (2)
- ;
- (3)
- ϕ is continuous at .
- (1)
- ψ is non-decreasing;
- (2)
- ;
- (3)
- For any sequence converging to 0, the sequence also converges to 0, where denotes the n-th iterate of ψ.
- )
- For any sequence , if and only if ;
- )
- For any sequence , if and only if .
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
GNMS | Generalized Neutrosophic Netric Spaces |
CTN | Continuous t-norm |
CTCN | Continuous t-conorm |
References
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
- Kramosil, I.; Michalek, J. Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11, 326–334. [Google Scholar]
- George, A.; Veeramani, P. On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64, 395–399. [Google Scholar] [CrossRef]
- Sun, G.; Yang, K. Generalized fuzzy metric spaces with properties. Res. J. Appl. Sci. Eng. Technol. 2010, 2, 673–678. [Google Scholar]
- Park, J.H. Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22, 1039–1046. [Google Scholar] [CrossRef]
- Jeyaraman, M.; Suganthi, M.; Shatanawi, W. Common fixed point theorems in intuitionistic generalized fuzzy cone metric spaces. Mathematics 2020, 8, 1212. [Google Scholar] [CrossRef]
- Smarandache, F. Neutrosophy: Neutrosophic Probability, Set, and Logic; ProQuest Information and Learning: Ann Arbor, MI, USA, 1998. [Google Scholar]
- Kirisci, M.; Simsek, N. Neutrosophic metric spaces. Math. Sci. 2020, 14, 241–248. [Google Scholar] [CrossRef]
- Sowndrarajan, S.; Jeyaraman, M.; Smarandache, F. Fixed point results for contraction theorems in neutrosophic metric spaces. Neutrosoph. Sets Syst. 2020, 36, 23. [Google Scholar]
- Farokhzad Rostami, R. Fixed point theorems in non-Archimedean G-fuzzy metric spaces with new type contractive mappings. Int. J. Nonlinear Anal. Appl. 2024, 15, 49–60. [Google Scholar] [CrossRef]
- Johnsy, J.; Jeyaraman, M. Fixed point theorems for (ψ-ϕ)-contractions in generalized neutrosophic metric spaces. Bull. Math. Anal. Appl. 2024, 16, 13–25. [Google Scholar]
- Akram, M.; Ishtiaq, U.; Ahmad, K.; Guran, L. Some generalized neutrosophic metric spaces and fixed point results with applications. Symmetry 2024, 16, 965. [Google Scholar] [CrossRef]
- Mihet, D. Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 2008, 159, 739–744. [Google Scholar] [CrossRef]
- Al-Khaleel, M.; Al-Sharif, S.; AlAhmad, R. On Cyclic Contractive Mappings of Kannan and Chatterjea Type in Generalized Metric Spaces. Mathematics 2023, 11, 890. [Google Scholar] [CrossRef]
- Gupta, V.; Mani, N. Existence and uniqueness of fixed point in fuzzy metric spaces and its applications. In Advances in Intelligent Systems and Computing; Springer: Berlin/Heidelberg, Germany, 2014; Volume 236, pp. 217–224. [Google Scholar] [CrossRef]
- Wardowski, D. Fuzzy contractive mappings and fixed points in fuzzy metric space. Fuzzy Sets Syst. 2013, 222, 108–114. [Google Scholar] [CrossRef]
- Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3, 133–181. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Johnsy, J.A.; Jeyaraman, M.; Shukla, R. A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces. Symmetry 2024, 16, 1446. https://doi.org/10.3390/sym16111446
Johnsy JA, Jeyaraman M, Shukla R. A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces. Symmetry. 2024; 16(11):1446. https://doi.org/10.3390/sym16111446
Chicago/Turabian StyleJohnsy, Joseph Amalraj, Mathuraiveeran Jeyaraman, and Rahul Shukla. 2024. "A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces" Symmetry 16, no. 11: 1446. https://doi.org/10.3390/sym16111446
APA StyleJohnsy, J. A., Jeyaraman, M., & Shukla, R. (2024). A Symmetric View of Fixed-Point Results in Non-Archimedean Generalized Neutrosophic Metric Spaces. Symmetry, 16(11), 1446. https://doi.org/10.3390/sym16111446