Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
2.1. Proinov Contractions
- ()
- ψ is nondecreasing;
- ()
- for any ;
- ()
- for any .
2.2. Fuzzy Metric Spaces
- (GV-1)
- ;
- (GV-2)
- if, and only if, ;
- (GV-3)
- ;
- (GV-4)
- ;
- (GV-5)
- is a continuous function.
- -Cauchyif for all and all , there is such that for all ;
- -convergent to if for all and all , there is such that for all (in such a case, we write ).
3. Proinov-Type Fixed-Point Theory in Non-Archimedean Fuzzy Metric Spaces
- (Þ)
- φ is nondecreasing;
- (Þ)
- for any ;
- (Þ)
- for any ;
- (Þ)
- if is such that , then .
- If , then:In such a case, Property (Þ leads to:In particular,
- If , then (6) and Property (Þ ensure that:As is nondecreasing by (Þ, then:
- If , thenIn such a case, the assumption (Þ) guarantees that . In particular, ;
- If , then:
- (Þ)
- φ is nondecreasing;
- (Þ)
- for any ;
- (Þ)
- for any ;
- (Þ)
- .
4. Conclusions and Open Problems
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Bataihah, A.; Shatanawi, W.; Qawasmeh, T.; Hatamleh, R. On H-simulation functions and fixed-point results in the setting of ωt-distance mappings with application on matrix equations. Mathematics 2020, 8, 837. [Google Scholar] [CrossRef]
- Proinov, P.D. Fixed point theorems for generalized contractive mappings in metric spaces. J. Fixed Point Theory Appl. 2020, 22, 21. [Google Scholar] [CrossRef]
- Olaru, I.M.; Secelean, N.A. A new approach of some contractive mappings on metric spaces. Mathematics 2021, 9, 1433. [Google Scholar] [CrossRef]
- Alqahtani, B.; Alzaid, S.S.; Fulga, A.; Roldán López de Hierro, A.F. Proinov type contractions on dislocated b-metric spaces. Adv. Differ. Equ. 2021, 2021, 164. [Google Scholar] [CrossRef]
- Alghamdi, M.A.; Gulyaz-Ozyurt, S.; Fulga, A. Fixed Points of Proinov E-Contractions. Symmetry 2021, 13, 962. [Google Scholar] [CrossRef]
- Alamri, B.A.S.; Agarwal, R.P.; Ahmad, J. Some new fixed Point theorems in b-metric spaces with application. Mathematics 2020, 8, 725. [Google Scholar] [CrossRef]
- Karapınar, E.; Fulga, A.; Petruşel, A. On Istrăţescu type contractions in b-metric spaces. Mathematics 2020, 8, 388. [Google Scholar] [CrossRef] [Green Version]
- Menger, K. Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28, 535–537. [Google Scholar] [CrossRef] [Green Version]
- Kaleva, O.; Seikkala, S. On fuzzy metric spaces. Fuzzy Sets Syst. 1984, 12, 215–229. [Google Scholar] [CrossRef]
- Schweizer, B.; Sklar, A. Probabilistic Metric Spaces; Dover Publications: New York, NY, USA, 2005. [Google Scholar]
- Kramosil, I.; Michálek, J. Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11, 336–344. [Google Scholar]
- Roldán López de Hierro, A.F.; de la Sen, M.; Roldán López de Hierro, C.; Martínez-Moreno, J. An approach version of fuzzy metric spaces including an ad hoc fixed point theorem. Fixed Point Theory Appl. 2015, 33, 1–23. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Shahzad, N. Fuzzy (and probabilistic) RS-metric spaces and related fixed-point theory. J. Nonlinear Convex Anal 2017, 17, 2365–2383. [Google Scholar]
- George, A.; Veeramani, P. On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64, 395–399. [Google Scholar] [CrossRef] [Green Version]
- Al-Mezel, S.A.; Alsulami, H.H.; Karapınar, E.; Roldán López de Hierro, A.F. Discussion on “multidimensional coincidence points” via recent publications. Abstr. Appl. Anal. 2014, 2014, 287492. [Google Scholar] [CrossRef]
- Altun, I.; Miheţ, D. Ordered non-Archimedean fuzzy metric spaces and some fixed-point results. Fixed Point Theory Appl. 2010, 2010, 782680. [Google Scholar] [CrossRef] [Green Version]
- Gregori, V.; Sapena, A. On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2020, 125, 245–252. [Google Scholar] [CrossRef]
- Karapınar, E. Revisiting the Kannan type contractions via interpolation. Adv. Theory Nonlinear Anal. Appl. 2018, 2, 85–87. [Google Scholar] [CrossRef] [Green Version]
- Miheţ, D. Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 2008, 159, 739–744. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Karapınar, E.; Manro, S. Some new fixed point theorems in fuzzy metric spaces. J. Intell. Fuzzy Syst. 2014, 27, 2257–2264. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Shahzad, N. Ample spectrum contractions and related fixed point theorems. Mathematics 2019, 7, 1033. [Google Scholar] [CrossRef] [Green Version]
- Shahzad, N.; Roldán López de Hierro, A.F.; Khojasteh, F. Some new fixed point theorems under (A,S)-contractivity conditions. RACSAM Rev. R. Acad. A 2017, 111, 307–324. [Google Scholar]
- Suganthi, M.; Mathuraiveeran, M.J. Common fixed point theorems in M-fuzzy cone metric space. Results Nonlinear Anal. 2021, 4, 33–46. [Google Scholar] [CrossRef]
- Roldán, A.; Martínez-Moreno, J.; Roldán, C. On interrelationships between fuzzy metric structures. Iran. J. Fuzzy Syst. 2010, 10, 133–150. [Google Scholar]
- Alfonso, G.; Roldán López de Hierro, A.F.; Roldán, C. A fuzzy regression model based on finite fuzzy numbers and its application to real-world financial data. J. Comput. Appl. Math. 2017, 318, 47–58. [Google Scholar] [CrossRef]
- Chen, L.H.; Hsueh, C.-C. Fuzzy regression models using the least-squares method based on the concept of distance. IEEE Trans. Fuzzy Syst. 2009, 17, 1259–1272. [Google Scholar] [CrossRef]
- Hesamian, G.; Akbari, M.G. A fuzzy additive regression model with exact predictors and fuzzy responses. Appl. Soft Comput. J. 2020, 95, 106507. [Google Scholar] [CrossRef]
- Maturo, F.; Hošková-Mayerová, Š. Fuzzy regression models and alternative operations for economic and social sciences. In Recent Trends in Social Systems: Quantitative Theories and Quantitative Models. Studies in Systems, Decision and Control; Maturo, A., Hošková-Mayerová, Š., Soitu, D.T., Kacprzyk, J., Eds.; Springer: Cham, Switzerland, 2017; Volume 66, pp. 235–247. [Google Scholar]
- Roldán, C.; Roldán, A.; Martínez-Moreno, J. A fuzzy regression model based on distances and random variables with crisp input and fuzzy output data: A case study in biomass production. Soft Comput. 2012, 16, 785–795. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Roldán López de Hierro, C.; Martínez-Moreno, J.; Aguilar Peña, C. Estimation of a fuzzy regression model using fuzzy distances. IEEE Trans. Fuzzy Syst. 2016, 24, 344–359. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Karapınar, E.; O’Regan, D.; Roldán López de Hierro, A.F. Fixed Point Theory in Metric Type Spaces; Springer International Publishing: Berlin, Germany, 2015. [Google Scholar]
- Roldán López de Hierro, A.F.; Shahzad, N. Fixed point theorems by combining Jleli and Samet’s, and Branciari’s inequalities. J. Nonlinear Sci. Appl. 2016, 9, 3822–3849. [Google Scholar]
- Klement, E.P.; Mesiar, R.; Pap, E. Triangular Norms; Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000. [Google Scholar]
- Grabiec, M. Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27, 385–389. [Google Scholar] [CrossRef]
- Istrăţescu, V. An Introduction to Theory of Probabilistic Metric Spaces with Applications; Tehnică, Bucureşti: Bucharest, Romania, 1974. [Google Scholar]
- Roldán López de Hierro, A.F.; Karapınar, E.; Fulga, A. Multiparametric contractions and related Hardy-Roger type fixed point theorems. Mathematics 2020, 8, 957. [Google Scholar] [CrossRef]
- Roldán López de Hierro, A.F.; Karapınar, E.; Shahzad, N. Fuzzy ample spectrum contractions in (more general than) non-Archimedean fuzzy metric spaces. arXiv 2021, arXiv:2104.09155. [Google Scholar]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 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
Roldán López de Hierro, A.F.; Fulga, A.; Karapınar, E.; Shahzad, N. Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces. Mathematics 2021, 9, 1594. https://doi.org/10.3390/math9141594
Roldán López de Hierro AF, Fulga A, Karapınar E, Shahzad N. Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces. Mathematics. 2021; 9(14):1594. https://doi.org/10.3390/math9141594
Chicago/Turabian StyleRoldán López de Hierro, Antonio Francisco, Andreea Fulga, Erdal Karapınar, and Naseer Shahzad. 2021. "Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces" Mathematics 9, no. 14: 1594. https://doi.org/10.3390/math9141594
APA StyleRoldán López de Hierro, A. F., Fulga, A., Karapınar, E., & Shahzad, N. (2021). Proinov-Type Fixed-Point Results in Non-Archimedean Fuzzy Metric Spaces. Mathematics, 9(14), 1594. https://doi.org/10.3390/math9141594