Kybernetika 52 no. 6, 929-942, 2016

Caristi's fixed point theorem and its equivalences in fuzzy metric spaces

Naser Abbasi and Hamid Mottaghi GolshanDOI: 10.14736/kyb-2016-6-0929


In this article, we extend Caristi's fixed point theorem, Ekeland's variational principle and Takahashi's maximization theorem to fuzzy metric spaces in the sense of George and Veeramani [A. George , P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems. 64 (1994) 395-399]. Further, a direct simple proof of the equivalences among these theorems is provided.


fuzzy metric space, Ekeland variational principle, Caristi's fixed point theorem, Takahashi's maximization theorem


47H10, 58E30


  1. I. Altun and D. Mihet: Ordered non-archimedean fuzzy metric spaces and some fixed point results. Fixed Point Theory Appl. 2010, Art. ID 782680, 11 pp.   DOI:10.1155/2010/782680
  2. J.-P. Aubin: Optima and equilibria. An introduction to nonlinear analysis. Translated from the French by Stephen Wilson. Second edition. Springer-Verlag, Graduate Texts in Mathematics 149, Berlin 1998.   CrossRef
  3. J. S. Bae, E. W. Cho and S. H. Yeom: A generalization of the {C}aristi-{K}irk fixed point theorem and its applications to mapping theorems. J. Korean Math. Soc. 31 (1994), 1, 29-48.   CrossRef
  4. S. Banach: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 3 (1922), 1, 133-181.   CrossRef
  5. A. Brøndsted: Fixed points and partial orders. Proc. Amer. Math. Soc. 60 (1976), 365-366.   DOI:10.1090/s0002-9939-1976-0417867-x
  6. F. E. Browder: On a theorem of {C}aristi and {K}irk. In: Proc. Sem. Fixed point theory and its applications {D}alhousie {U}niv., {H}alifax, 1975), Academic Press, New York 1976, pp. 23-27.   CrossRef
  7. J. Caristi: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215 (1976), 241-251.   DOI:10.1090/s0002-9947-1976-0394329-4
  8. J. Caristi and W. A. Kirk: Geometric fixed point theory and inwardness conditions. In: Proc. Conf. The geometry of metric and linear spaces, {M}ichigan {S}tate {U}niv., {E}ast {L}ansing 1974), Lecture Notes in Math. 490, Springer, Berlin 1975, pp. 74-83.   DOI:10.1007/bfb0081133
  9. S. S. Chang and Q. Luo: Caristi's fixed point theorem for fuzzy mappings and {E}keland's variational principle. Fuzzy Sets and Systems 64 (1994), 1, 119-125.   DOI:10.1016/0165-0114(94)90014-0
  10. I. Ekeland: Sur les problèmes variationnels. C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A1057-A1059.   CrossRef
  11. I. Ekeland: On the variational principle. J. Math. Anal. Appl. 47 (1974), 324-353.   DOI:10.1016/0022-247x(74)90025-0
  12. I. Ekeland: Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979), 3, 443-474.   DOI:10.1090/s0273-0979-1979-14595-6
  13. A. George and P. Veeramani: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64 (1994), 3, 395-399.   DOI:10.1016/0165-0114(94)90162-7
  14. A. George and P. Veeramani: Some theorems in fuzzy metric spaces. J. Fuzzy Math. 3 (1995), 4, 933-940.   CrossRef
  15. M. Grabiec: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 27 (1988), 3, 385-389.   DOI:10.1016/0165-0114(88)90064-4
  16. V. Gregori, J.-J. Mi{ñ}ana and S. Morillas: Some questions in fuzzy metric spaces. Fuzzy Sets and Systems 204 (2012), 71-85.   DOI:10.1016/j.fss.2011.12.008
  17. V. Gregori, S. Morillas and A. Sapena: On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems 161 (2010), 16, 2193-2205.   DOI:10.1016/j.fss.2010.03.013
  18. V. Gregori, S. Morillas and A. Sapena: Examples of fuzzy metrics and applications. Fuzzy Sets and Systems 170 (2011), 95-111.   DOI:10.1016/j.fss.2010.10.019
  19. V. Gregori and S. Romaguera: On completion of fuzzy metric spaces. Fuzzy Sets and Systems 130 (2002), 3, 399-404.   DOI:10.1016/s0165-0114(02)00115-x
  20. V. Gregori and S. Romaguera: Characterizing completable fuzzy metric spaces. Fuzzy sets and systems 144 (2004), 3, 411-420.   DOI:10.1016/s0165-0114(03)00161-1
  21. O. Had{ž}i{ć} and E. Pap: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic Publishers, Mathematics and its Applications 536, Dordrecht 2001.   DOI:10.1007/978-94-017-1560-7
  22. J. S. Jung, Y. J. Cho, S. M. Kang and S.-S. Chang: Coincidence theorems for set-valued mappings and {E}keland's variational principle in fuzzy metric spaces. Fuzzy Sets and Systems 79 (1996), 2, 239-250.   DOI:10.1016/0165-0114(95)00084-4
  23. J. S. Jung, Y. J. Cho and J. K. Kim: Minimization theorems for fixed point theorems in fuzzy metric spaces and applications. Fuzzy Sets and Systems 61 (1994), 2, 199-207.   DOI:10.1016/0165-0114(94)90234-8
  24. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Trends in Logic--Studia Logica Library 8, Dordrecht 2000.   DOI:10.1007/978-94-015-9540-7
  25. I. Kramosil and J. Mich{á}lek: Fuzzy metrics and statistical metric spaces. Kybernetika 11 (1975), 5, 336-344.   CrossRef
  26. G. M. Lee, B. S. Lee, J. S. Jung and S.-S. Chang: Minimization theorems and fixed point theorems in generating spaces of quasi-metric family. Fuzzy Sets and Systems 101 (1999), 1, 143-152.   DOI:10.1016/s0165-0114(97)00034-1
  27. K. Menger: Statistical metrics. Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 535-537.   DOI:10.1073/pnas.28.12.535
  28. V. Radu: Some remarks on the probabilistic contractions on fuzzy {M}enger spaces. Automat. Comput. Appl. Math. 11 (2003), 1, 125-131.   CrossRef
  29. J. Rodríguez-L{ó}pez and S. Romaguera: The {H}ausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 147 (2004), 2, 273 -283.   DOI:10.1016/j.fss.2003.09.007
  30. B. Schweizer and A. Sklar: Statistical metric spaces. Pacific J. Math. 10 (1960), 313-334.   DOI:10.2140/pjm.1960.10.313
  31. B. Schweizer and A. Sklar: Probabilistic metric spaces    CrossRef
  32. T. Suzuki: On {D}owning-{K}irk's theorem. J. Math. Anal. Appl. 286 (2003), 2, 453-458.   DOI:10.1016/s0022-247x(03)00470-0
  33. T. Suzuki: Generalized {C}aristi's fixed point theorems by {B}ae and others. J. Math. Anal. Appl. 302 (2005), 2, 502-508.   DOI:10.1016/j.jmaa.2004.08.019
  34. T. Suzuki and W. Takahashi: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 8 (1997), 2, 371-382.   CrossRef
  35. W. Takahashi: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Fixed Point Theory and Aplications {M}arseille, 1989), Pitman Res. Notes Math. Ser. 252, Longman Sci. Tech., Harlow 1991, pp. 397-406.   CrossRef
  36. W. Takahashi: Nonlinear functional analysis. Yokohama Publishers, Yokohama 2000.   CrossRef
  37. J. Zhu, C.-K. Zhong and G.-P. Wang: An extension of ekeland's variational principle in fuzzy metric space and its applications. Fuzzy Sets and Systems 108 (1999), 3, 353-363.   DOI:10.1016/s0165-0114(97)00333-3