1.30

Impact Factor

2.10

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# Contraction Mapping Theorem in Partial Fuzzy Metric Spaces

1Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq
2Department of Mathematics, College of Education, Al-Mustansiriyah University, Baghdad, Iraq

Accepted: September 6, 2021
Publication Date: September 29, 2021

Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

## ABSTRACT

Contraction mapping theorem and its application is often encountered different fields of pure and applied mathematics. In this work, three aims are achieved, which are first of introducing and presenting the basic and fundamental necessary concepts for partial fuzzy metric spaces. The second aim is to state and prove two fuzzy contraction mapping theorems in fuzzy partial metric spaces based on the contractive mapping theorem by using the suggested metric function that is defined over fuzzy points rather than fuzzy sets. Also, we present in this paper, as the third aim, the relationship between the usual distance function defined in fuzzy metric spaces and the distance function defined over partial metric space, which are defined using fuzzy points.

Keywords: Partial fuzzy metric space, Contraction mapping theorem, Fuzzy points, Contraction mapping

## REFERENCES

1. [1] L. A. Zadeh, (1965) “Information and control" Fuzzy sets 8(3): 338–353.
2. [2] I. Kramosil and J. Michálek, (1975) “Fuzzy metrics and statistical metric spaces" Kybernetika 11(5): 336–344.
3. [3] A. George and P. Veeramani, (1997) “On some results of analysis for fuzzy metric spaces" Fuzzy sets and systems 90(3): 365–368.
4. [4] A. George and P. Veeramani, (1994) “On some results in fuzzy metric spaces" Fuzzy sets and systems 64(3): 395–399.
5. [5] J. Rodríguez-López and S. Romaguera, (2004) “The Hausdorff fuzzy metric on compact sets" Fuzzy sets and systems 147(2): 273–283.
6. [6] V. Gregori and A. Sapena, (2002) “On fixed-point theorems in fuzzy metric spaces" Fuzzy sets and systems 125(2): 245–252.
7. [7] D. Dey and M. Saha, (2013) “Partial cone metric space and some fixed point theorems" TWMS Journal of Applied and Engineering Mathematics 3(1): 1–9.
8. [8] M. Abbas, I. Altun, and D. Gopal, (2009) “Common fixed point theorems for non compatible mappings in fuzzy metric spaces" Bull. Math. Anal. Appl 1(2): 47–56.
9. [9] F. Kiany and A. Amini-Harandi, (2011) “Fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces" Fixed Point Theory and Applications 2011(1): 1–9.
10. [10] M. A. Alghamdi, N. Shahzad, and O. Valero, (2012) “On fixed point theory in partial metric spaces" Fixed Point Theory and Applications 2012(1): 1–25.
11. [11] D. Paesano and C. Vetro, (2014) “Multi-valued Fcontractions in O-complete partial metric spaces with application to Volterra type integral equation" Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 108(2): 1005–1020.
12. [12] R. Pant, R. Shukla, H. Nashine, and R. Panicker, (2017) “Some new fixed point theorems in partial metric spaces with applications" Journal of Function Spaces 2017:
13. [13] S. Shukla, D. Gopal, and A.-F. Roldán-López-de- Hierro, (2016) “Some fixed point theorems in 1-Mcomplete fuzzy metric-like spaces" International Journal of General Systems 45(7-8): 815–829.
14. [14] M. Abbas, F. Lael, and N. Saleem, (2020) “Fuzzy bmetric spaces: fixed point results for y-contraction correspondences and their application" Axioms 9(2): 1–12.
15. [15] N. Saleem, H. I¸sık, S. Furqan, and C. Park, (2021) “Fuzzy double controlled metric spaces and related results" Journal of Intelligent & Fuzzy Systems (Preprint): 1–9.
16. [16] S. Furqan, H. I¸sık, and N. Saleem, (2021) “Fuzzy triple controlled metric spaces and related fixed point results" Journal of Function Spaces 2021:
17. [17] B. Mohammadi, A. Hussain, V. Parvaneh, N. Saleem, and R. J. Shahkoohi, (2021) “Fixed Point Results for Generalized Fuzzy Contractive Mappings in Fuzzy Metric Spaces with Application in Integral Equations" Journal of Mathematics 2021:
18. [18] S. G. Matthews, (1994) “Partial metric topology" Annals of the New York Academy of Sciences-Paper Edition 728: 183–197.
19. [19] F. J. Amer. “Fuzzy partial metric spaces”. In: Computational Analysis. Springer, 2016, 153–161.
20. [20] K. S. Eke, J. G. Oghonyon, and B. Davvaz, (2018) “Some fixed point theorems for contractive maps in fuzzy G-partial metric spaces" International Journal of Mechanical Engineering and Technology (IJMET) 9(8):0976–6359.
21. [21] S. J. O’NEILL, (1996) “Partial metrics, valuations, and domain theory" Annals of the New York Academy of Sciences 806(1): 304–315.
22. [22] S. Sedghi, N. Shobkolaei, and I. Altun, (2015) “Partial fuzzy metric space and some fixed point results" Communications in Mathematics 23(2): 131–142.
23. [23] Z. Deng, (1982) “Fuzzy pseudo-metric spaces" Journal of Mathematical Analysis and Applications 86(1):74–95.
24. [24] H. M. Wali, (2018) “Some Results of Principal Fuzzy Metric Spaces and Their Completeness" Al-Nahrain Journal of Science (1): 119–122.
25. [25] V. Gregori, J.-J. Miñana, and D. Miravet, (2020) “A Duality Relationship Between Fuzzy Partial Metrics and Fuzzy Quasi-Metrics" Mathematics 8(9): 1575.
26. [26] V. Gregori, J.-J. Miñana, and D. Miravet, (2019) “Fuzzy partial metric spaces" International Journal of General Systems 48(3): 260–279.
27. [27] I. Beg, D. Gopal, T. Dosenovic, and D. Rakic, (2018) “a-Type fuzzy H-contractive mappings in fuzzy metric spaces" Fixed Point Theory 19(2): 463–474.

2.1
2023CiteScore

69th percentile