The notion of fuzzy logic was introduced by Zadeh. Unlike traditional logic theory, where an element either belongs to the set or does not, in fuzzy logic, the affiliation of the element to the set is expressed as a number from the interval [0, 1]. The study of the theory of fuzzy sets was prompted by the presence of uncertainty as an essential part of real-world problems, leading Zadeh to address the problem of indeterminacy. The theory of a fixed point in fuzzy metric spaces can be viewed in different ways, one of which involves the use of fuzzy logic. Fuzzy metric spaces, which are specific types of topological spaces with pleasing ”geometric” characteristics, possess a number of appealing properties and are commonly used in both pure and applied sciences. Metric spaces and their various generalizations frequently occur in computer science applications. For this reason, a new space called a Pompeiu-Hausdorff fuzzy b-metric space is constructed in this paper. In this space, some new fixed point results are also formulated and proven. Additionally, a general common fixed point theorem for a pair of multi-valued mappings in Pompeiu-Hausdorff fuzzy b-metric spaces is investigated. The findings obtained in fuzzy metric spaces, such as those discussed in Remark 3.1, are generalized by the results in this paper, and additional specific findings are produced and supported by examples. The study of denotational semantics and their applications in control theory using fuzzy b-metric spaces and Pompeiu-Hausdorff fuzzy b-metric spaces will be an important next step.
| Published in | Applied and Computational Mathematics (Volume 13, Issue 5) |
| DOI | 10.11648/j.acm.20241305.11 |
| Page(s) | 118-129 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Fuzzy Metric Space, Fuzzy b-metric Space, t-norm, Fixed Point, Implicit Relation
| [1] | A. Aghajani, M. Abbas, J. R. Roshan,Common Fixed Point of Generalized Weak Contractive Mappings in Partially Ordered b-metric Spaces, Math. Slovaca 64, No. 4, 941-960, (2014). https://doi.org/10.2478 s12175-014- 0250-6 |
| [2] | S Aleksić, Z. D. Mitrović, S Radenović, Picard sequences in b-metric spaces, Fixed point theory, 21 (2020), No. 1, 34-46. https://doi.org/10.24193/fpt-ro.2020.1.03 http://www.math.ubbcluj.ro/nodeacj/sfptcj.html |
| [3] | M. U. Ali, T. Kamran, M. Postolache, Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem. Nonlinear Anal. Modelling Control 22, No. 1, 17-30, (2017). http://dx.doi.org/10.15388/NA.2017.1.2 |
| [4] | I. A. Bakhtin, The contraction mapping principle in almost metric spaces. 30. In Functional Analysis. Ul’yanovsk Gos. Ped. Inst., Ul’yanovsk, 26-37,(1989). |
| [5] | S. Czerwik, Nonlinear set valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fiz. Univ. Modena, 46, 263-276, (1998). |
| [6] | S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav, 1, 5-11, (1993). |
| [7] | M. Dahhouch, N. makran, B. Marzouki, A Generalized Fixed Point Theorem in Fuzzy b−Metric Spaces and Applications, Bol. Soc. Paran. Mat. v. 22 2, (2024). https://doi.org/10.5269/bspm.63276 |
| [8] | M. Dahhouch, N. Makran, B. Marzouki, A Common Fixed Point Of Multivalued Maps In Extended b-Metric Space With Application Volterra-Type Integral Inclusion. U.P.B. Sci. Bull., Series A, Vol. 84, Iss 4, (2022). |
| [9] | T. Došenović, A. Javaheri, S. Sedghi, N. Shobe, Coupled fixed point theorem in b-fuzzy metric spaces. Novi Sad J. Math. 47(1), 77-88, (2017). https://doi.org/10.30755/NSJOM.04361 |
| [10] | T. Došenović, D. Rakić, S. Radenović, B. Carić, Ćirić type nonunique fixed point theorems in the frame of fuzzy metric spaces, AIMS Mathematics, (2023). 8 (1): 2154- 2167, https://doi.org/10.3934/math |
| [11] | A. George, P. Veeramani, On some result in fuzzy metric space. Fuzzy Sets Syst. 64, 395-399, (1994). http://dx.doi.org/10.1016/0165-0114(94)90162-7 |
| [12] | A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365-368, (1997). http://dx.doi.org/10.1016/S0165- 0114(96)00207-2 |
| [13] | Z. Hassanzadeh, S. Sedghi, Relation between b-metric and fuzzy metric spaces. Math. Morav. 22(1), 55-63, (2018). https://doi.org/10.5937/MatMor1801055H. |
| [14] | I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11, 326-334, (1975). |
| [15] | N. Makran, A. El Haddouchi, B. Marzouki, A common fixed point of multi-valued maps in b-metric space. U.P.B. Sci. Bull., Series A, Vol. 82, Iss. 1, (2020). |
| [16] | N. Makran, A. El Haddouchi, B. Marzouki, A generalized common fixed points for multivalued mappings in Gb- metric spaces with an Application . U.P.B. Sci. Bull., Series A, Vol. 83, Iss. 1, (2021). |
| [17] | N. Makran, A. El Haddouchi, B. Marzouki, A Generalized Common Fixed Point of Multi-Valued Maps in b−metric Space, Bol. Soc. Paran. Mat. v. 1-9, (2021). https://doi.org/10.5269/bspm.51655 |
| [18] | N. Makran, O. Hammouti, S. Taarabti, A common fixed point result for multi-valued mappings in Hausdorff modularfuzzyb-metricspaceswithapplicationtointegral inclusions, Analysis (2024), https://doi.org/10.1515/anly- 2023-0081. |
| [19] | B. Marzouki, N. Makran, A. El Haddouchi , A generalized Common Fixed Point Theorem in Complex Valued b−Metric Spaces Bol. Soc. Paran. Mat. v 40, (2022). https://doi.org/10.5269/bspm.51616 |
| [20] | S. Sedghi, N. Shobe, Common fixed point theorem in b- fuzzy metric space. Nonlinear Funct. Anal. Appl. 17(3), 349-359, (2012). |
| [21] | S. Sedghi, N. Shobe, Common fixed point theorem for R-weakly commuting maps in b-fuzzy metric space. Nonlinear Funct. Anal. Appl. 19(2), 285-295, (2014). |
| [22] | Y. Shen, D. Qiu, W. Chen Fixed point theorems in fuzzy metric spaces. Applied Mathematics Letters 25, 138-141, (2012). https://doi.org/10.1016/j.aml.2011.08.002 |
APA Style
Makran, N. (2024). Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings. Applied and Computational Mathematics, 13(5), 118-129. https://doi.org/10.11648/j.acm.20241305.11
ACS Style
Makran, N. Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings. Appl. Comput. Math. 2024, 13(5), 118-129. doi: 10.11648/j.acm.20241305.11
AMA Style
Makran N. Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings. Appl Comput Math. 2024;13(5):118-129. doi: 10.11648/j.acm.20241305.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.acm.20241305.11,
author = {Noreddine Makran},
title = {Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings},
journal = {Applied and Computational Mathematics},
volume = {13},
number = {5},
pages = {118-129},
doi = {10.11648/j.acm.20241305.11},
url = {https://doi.org/10.11648/j.acm.20241305.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20241305.11},
abstract = {The notion of fuzzy logic was introduced by Zadeh. Unlike traditional logic theory, where an element either belongs to the set or does not, in fuzzy logic, the affiliation of the element to the set is expressed as a number from the interval [0, 1]. The study of the theory of fuzzy sets was prompted by the presence of uncertainty as an essential part of real-world problems, leading Zadeh to address the problem of indeterminacy. The theory of a fixed point in fuzzy metric spaces can be viewed in different ways, one of which involves the use of fuzzy logic. Fuzzy metric spaces, which are specific types of topological spaces with pleasing ”geometric” characteristics, possess a number of appealing properties and are commonly used in both pure and applied sciences. Metric spaces and their various generalizations frequently occur in computer science applications. For this reason, a new space called a Pompeiu-Hausdorff fuzzy b-metric space is constructed in this paper. In this space, some new fixed point results are also formulated and proven. Additionally, a general common fixed point theorem for a pair of multi-valued mappings in Pompeiu-Hausdorff fuzzy b-metric spaces is investigated. The findings obtained in fuzzy metric spaces, such as those discussed in Remark 3.1, are generalized by the results in this paper, and additional specific findings are produced and supported by examples. The study of denotational semantics and their applications in control theory using fuzzy b-metric spaces and Pompeiu-Hausdorff fuzzy b-metric spaces will be an important next step.},
year = {2024}
}
TY - JOUR T1 - Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings AU - Noreddine Makran Y1 - 2024/08/29 PY - 2024 N1 - https://doi.org/10.11648/j.acm.20241305.11 DO - 10.11648/j.acm.20241305.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 118 EP - 129 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20241305.11 AB - The notion of fuzzy logic was introduced by Zadeh. Unlike traditional logic theory, where an element either belongs to the set or does not, in fuzzy logic, the affiliation of the element to the set is expressed as a number from the interval [0, 1]. The study of the theory of fuzzy sets was prompted by the presence of uncertainty as an essential part of real-world problems, leading Zadeh to address the problem of indeterminacy. The theory of a fixed point in fuzzy metric spaces can be viewed in different ways, one of which involves the use of fuzzy logic. Fuzzy metric spaces, which are specific types of topological spaces with pleasing ”geometric” characteristics, possess a number of appealing properties and are commonly used in both pure and applied sciences. Metric spaces and their various generalizations frequently occur in computer science applications. For this reason, a new space called a Pompeiu-Hausdorff fuzzy b-metric space is constructed in this paper. In this space, some new fixed point results are also formulated and proven. Additionally, a general common fixed point theorem for a pair of multi-valued mappings in Pompeiu-Hausdorff fuzzy b-metric spaces is investigated. The findings obtained in fuzzy metric spaces, such as those discussed in Remark 3.1, are generalized by the results in this paper, and additional specific findings are produced and supported by examples. The study of denotational semantics and their applications in control theory using fuzzy b-metric spaces and Pompeiu-Hausdorff fuzzy b-metric spaces will be an important next step. VL - 13 IS - 5 ER -
Department of Mathematics, Mohammed Premier University, Oujda, Morocco
