Fuzzy Metric Spaces and Corresponding Fixed Point Theorems for Fuzzy Type Contraction
Musa Adeku Ibrahim
1
(
Department of Mathematics, Federal University Lokoja, Lokoja, Nigeria
)
Muhammed Raji
2
(
Department of Mathematics, Federal University Lokoja, Lokoja, Nigeria
)
Kamilu Rauf
3
(
Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Nigeria
)
Keywords: Fixed point, Fuzzy fixed point, Contractive type mapping, Hausdorff fuzzy metric space, Fuzzy mapping.,
Abstract :
In this paper, we present innovative concepts of fuzzy type contractions and leverage them to establish fixed point theorems for fuzzy mappings within the framework of fuzzy metric spaces. The results of this article are applied to multivalued mappings and fuzzy mappings for contractive fuzzy type mappings. Through illustrative examples, we showcase the practical applicability of our proposed notions and results, demonstrating their effectiveness in real-world scenarios.
[1] Azam A. Fuzzy fixed points of fuzzy mappings via a rational inequality. Hacettepe Journal of Mathematics and Statistics. 2011; 40(3): 421-431.
[2] Banach S. Sur les operations dans les ensembles abstraits et leur application aux equations itegrales. Fundamenta mathematicae. 1922; 3(1): 133-181. DOI: https://doi.org/10.4064/fm-3-1-133-181
[3] Boriceanu M. Strict fixed point theorems for multivalued operators in b-metric spaces. International Journal of Modern Mathematical Sciences. 2009; 4(3): 285-301.
[4] Butnariu D. Fixed point for fuzzy mapping.Fuzzy Sets and Systems. 1982; 7(2): 191-207. DOI: https://doi.org/10.1016/0165-0114(82)90049-5
[5] Ciri´c LB. A generalization of Banachs contraction principle. Proceedings of the American Mathematical Society. 1974; 45(2): 267-273. DOI: https://doi.org/10.2307/2040075
[6] Czerwik S. Contractive mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis. 1993; 1(1): 5-11.
[7] Heilpern S. Fuzzy mappings and fixed point theorem. Journal of Mathematical Analysis and Applications. 1981; 83(2): 566-569, DOI: https://doi.org/10.1016/0022-247X(81)90141-4
[8] Hitzler P, Seda AK. Dislocated topologies. Journal of Electrical Engineering. 2000; 51(12): 3-7.
[9] Hussain A, Ishtiaq U, Sulami HA. Fixed point results in fuzzy strong controlled metric spaces with an application to the domain words. Advances in Mathematical Physics. 2023; 1: 4350504, DOI: https://doi.org/10.1155/2023/4350504
[10] Hussain N, Roshan JR, Paravench V, Abbas M. Common fixed point results for weak contractive mappings in ordered dislocated b-metric space with applications. Journal of Inequalities and Applications. 2013; 486: 1-21. DOI: https://doi.org/10.1186/1029-242X-2013-486
[11] Hussain A, Sulami HA, Ishtiaq U. Some new aspects in the intuitionistic fuzzy and neutrosophic fixed point theory. Journal of Function Spaces. 2022; 1: 3138740. DOI: https://doi.org/10.1155/2022/3138740
[12] Ishtiaq U, Hussain A, Sulami HA. Certain new aspects in fuzzy fixed point theory. AIMS Mathematics. 2022; 7(5): 8558-8573. DOI: https://doi.org/10.3934/math.2022477
[13] Kanwal S, Ali A, Al Mazrooei A, Garcia GS. Existence of fuzzy fixed points of set-valued fuzzy mappings in metric and fuzzy metric spaces. AIMS Mathematics. 2023; 8(5): 10095-10112. https://doi.org/10.3934/math.2023511
[14] Kanwal S, Azam A, Shami FA. On coincidence theorem in intuitionistic fuzzy b-metric spaces with application. Journal of Function Spaces. 2022; 1: 5616824. DOI: https://doi.org/10.1155/2022/5616824
[15] Nadler BS. Multivalued contraction mappings. Pacific Journal of Mathematics. 1969; 30(2): 475-488. DOI: https://doi.org/10.2140/pjm.1969.30.475
[16] Phiangsungnoen S, Kumam P. Fuzzy fixed point theorems for multivalued fuzzy contractions in bmetric spaces. Journal of Nonlinear Sciences and Applications (JNSA). 2015; 8(1): 55-63. DOI: http://dx.doi.org/10.22436/jnsa.008.01.07
[17] Phiangsungnoen S, Sintunavarat W, Kumam P. Common α-fuzzy fixed point theorems for fuzzy mappings via βF-admissible pair. Journal of Intelligent &Fuzzy Systems. 2014; 27(5): 2463-2472. DOI: https://doi.org/10.1186/s40467-014-0020-6
[18] Raji M, Ibrahim MA. Fixed point theorems for fuzzy contractions mappings in a dislocated bmetric spaces with applications. Annals of Mathematics and Computer Science. 2024; 21: 1-13. DOI: https://doi.org/10.56947/amcs.v21.233
[19] Raji M, Ibrahim MA, Rauf K, Kehinde R. Common fixed point results for fuzzy F-contractive mappings in a dislocated metric spaces with application. Qeios, (2024), 7, 1-21; DOI:
https://doi.org/10.32388/SV98CN
[20] Rasham T, Kutbi MA, Hussain A, Chandok S. Fuzzy dominated nonlinear operators with applications. Journal of Intelligent & Fuzzy Systems. 2024; 1-15. https://content.iospress.com/articles/journalof-intelligent-and-fuzzy-systems/ifs238250
[21] Rasham T, Saeed F, Agarwal RP, Hussain A, Felhi A. Symmetrical hybrid coupled fuzzy fixed-point results on closed ball in fuzzy metric space with applications. Symmetry. 2023; 15(1): 30. DOI: https://doi.org/10.3390/sym15010030
[22] Rashid M, Shahzad A, Azam A. Fixed point theorems for L-fuzzy mappings in quasi-pseudo metric spaces. Journal of Intelligent & Fuzzy Systems. 2017; 32(1): 499-507. DOI: http://doi.org/10.3233/JIFS152261
[23] Shahzad A, Shoaib A, Khammahawong K, Kumam P. New Ciric type rational fuzzy F-contraction for common fixed points. In: Beyond Traditional Probabilistic Methods in Economics 2. Springer International Publishing; 2019. 215-229. DOI: https://doi.org/10.1007/978-3-030-04200-417
[24] Shahzad A, Shoaib A, Mahmood Q. Fixed point theorems for fuzzy mappings in b- metric space. Italian Journal of Pure and Applied Mathematics. 2017; 38: 419-427.
[25] Shoaib A, Kumam P, Shahzad A, Phiangsungnoen S, Mahmood Q. Fixed point results for fuzzy mappings in a b-metric space. Fixed Point Theory and Applications. 2018; 2: 1-12. DOI: https://doi.org/10.1186/s13663-017-0626-8
[26] Weiss MD. Fixed points and induced fuzzy topologies for fuzzy sets. Journal of Mathematical Analysis and Applications. 1975; 50(1): 142-150. DOI: https://doi.org/10.1016/0022-247X(75)90044-X
[27] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 338-353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X