Pompeiu-Hausdorff Fuzzy <i>b</i>-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings

Noreddine Makran

doi:doi:10.11648/j.acm.20241305.11

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Pompeiu-Hausdorff Fuzzy b-metric Spaces Are Associated with a Common Fixed Point and Multivalued Mappings

Noreddine Makran^*

Published in Applied and Computational Mathematics (Volume 13, Issue 5)

Received: 14 July 2024 Accepted: 14 August 2024 Published: 29 August 2024

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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.

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

Keywords

Fuzzy Metric Space, Fuzzy b-metric Space, t-norm, Fixed Point, Implicit Relation

References

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[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).
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[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
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[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).
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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

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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

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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

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@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}
}

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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
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Author Information

Noreddine Makran

Department of Mathematics, Mohammed Premier University, Oujda, Morocco

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http://orcid.org/0000-0001-8131-6733

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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

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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

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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

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@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}
}

Copy | Download