Kybernetika 60 no. 3, 394-411, 2024

Asymptotic fuzzy contractive mappings in fuzzy metric spaces

Dhananjay Gopal, Juan Martínez-Moreno and Rosana Rodríguez-LópezDOI: 10.14736/kyb-2024-3-0394

Abstract:

Fixed point theory in fuzzy metric spaces has grown to become an intensive field of research. However, due to the complexity involved in the nature of fuzzy metrics, the authors need to develop innovative machinery to establish new fixed point theorems in such kind of spaces. In this paper, we propose the concepts of asymptotic fuzzy $\psi$-contractive and asymptotic fuzzy Meir--Keeler mappings, and describe some new machinery by which the corresponding fixed point theorems are proved. In this sense, the techniques used for the proofs in Section 5 are completely new.

Keywords:

fuzzy metric space, fixed point, asymptotic fuzzy $\psi $-contractive mapping, asymptotic fuzzy Meir-Keeler mapping

Classification:

54H25, 47H10

paper.pdf

References:

  1. N. Abbasi and H. M. Golshan: Caristi's fixed point theorem and its equivalences in fuzzy metric spaces. Kybernetika 52 (2016), 6, 929-942.   DOI:10.14736/kyb-2016-6-0929
  2. D. W. Boyd and J. S. W. Wong: On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458-464.   DOI:10.1090/S0002-9939-1969-0239559-9
  3. A. George and P. Veeramani: On some results in fuzzy metric spaces. Fuzzy Sets Systems 64 (1994), 395-399.   DOI:10.1016/0165-0114(94)90162-7
  4. D. Gopal, M. Abbas and M. Imdad: $\psi$-weak contractions in fuzzy metric spaces. Iranian J. Fuzzy Syst. 8(5) (2011), 141-148.   CrossRef
  5. D. Gopal and J. Martínez-Moreno: Suzuki type fuzzy $\mathcal{Z}$-contractive mappings and fixed points in fuzzy metric spaces. Kybernetika 57 (2021), 6, 908-921.   DOI:10.14736/kyb-2021-6-0908
  6. D. Gopal and C. Vetro: Some new fixed point theorems in fuzzy metric spaces. Iranian J. Fuzzy Syst. 11(3) (2014), 95-107.   CrossRef
  7. M. Grabiec: Fixed points in fuzzy metric spaces. Fuzzy Sets Systems 27 (1988), 385-389.   DOI:10.1016/0165-0114(88)90064-4
  8. V. Gregori and J. J. Miñana: On fuzzy $\psi$-contractive mappings. Fuzzy Sets Systems 300 (2016), 93-101.   DOI:10.1016/j.fss.2015.12.010
  9. V. Gregori, J. J. Miñana, B. Roig and A. Sapena: A characterization of $p$ complete fuzzy metric spaces. Fuzzy Sets Systems 444 (2022), 144-155.   DOI:10.1016/j.fss.2021.12.001
  10. V. Gregori and A. Sapena: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets Systems 125 (2002), 245-252.   DOI:10.1016/S0165-0114(00)00088-9
  11. O. Hadžić and E. Pap: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic Publishers, Dordrecht 2001.   CrossRef
  12. J. Jachymski and I. Jóźwik: On Kirk's asymptotic contractions. J. Math. Anal. Appl. 300 (2004), 147-1592.   DOI:10.1016/j.jmaa.2004.06.037
  13. W. A. Kirk: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 277 (2003), 645-650.   DOI:10.1016/S0022-247X(02)00612-1
  14. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer, Dordrecht 2000.   CrossRef
  15. I. Kramosil and J. Michálek: Fuzzy metric and statistical metric spaces. Kybernetica 15 (1975), 326-334.   CrossRef
  16. S. Leader: Uniformly contractive fixed points in compact metric spaces. Proc. Amer. Math. Soc. 86 (1982), 153-158.   DOI:10.1090/S0002-9939-1982-0663887-2
  17. T. Lindstrom and D. A. Ross: A nonstandard approach to asymptotic fixed point theorems. J. Fixed Point Theory Appl. 25 (2023), 35.   DOI:10.1007/s11784-022-01028-6
  18. J. Martínez-Moreno, D. Gopal, V. Rakoćević and A. S. Ranadive: Caristi type mappings and characterization of completeness of Archimedean type fuzzy metric spaces. Adv. Computat. Intelligence 2 (2022), 1-7.   DOI:10.1007/s43674-021-00014-8
  19. D. Miheţ: Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Systems 159 (2008), 739-744.   DOI:10.1016/j.fss.2007.07.006
  20. J. J. Miñana, A. Sostak and O. Valero: On metrization of fuzzy metrics and application to fixed point theory. Fuzzy Sets Systems 468 (2023), 108625.   DOI:10.1016/j.fss.2023.108625
  21. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. Elsevier, New York 1983.   CrossRef
  22. A. Shoaib and K. Khaliq: Fixed-point results for generalized contraction in K-sequentially complete ordered dislocated fuzzy quasimetric spaces. Fixed Point Theory Algor. Sci. Engrg. 27 (2022), 1-22.   DOI:10.1186/s13663-022-00737-4
  23. A. Shoaib, A. Azam and A. Shahzad: Common fixed point results for the family of multivalued mappings satisfying contractions on a sequence in Hausdorff fuzzy metric spaces. J. Comput. Anal. Appl. 24 (2018), 692-699.   DOI:10.1007/s00041-017-9535-9
  24. S. Shukla, D. Gopal and W. Sintunavarat: A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets Systems 350 (2018), 85-95.   DOI:10.1016/j.fss.2017.10.001
  25. T. Suzuki: Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal. 64 (2006), 971-978.   DOI:10.1016/j.na.2005.04.054
  26. D. Zheng and P. Wang: Meir-Keeler theorems in fuzzy metric spaces. Fuzzy Sets Systems 370 (2019), 120-128.   DOI:10.1016/j.fss.2018.08.014