Kybernetika 57 no. 1, 46-59, 2021

Caristi's fixed point theorem in probabilistic metric spaces

Kianoush Fathi Vajargah, Hamid Mottaghi Golshan and Abbas Arjomand FarDOI: 10.14736/kyb-2021-1-0046


In this work, we define a partial order on probabilistic metric spaces and establish some new Caristi's fixed point theorems and Ekeland's variational principle for the class of (right) continuous and Archimedean t-norms. As an application, a partial answer to Kirk's problem in metric spaces is given.


probabilistic metric space, Caristi's fixed point, Archimedean t-norm


47H10, 58E30


  1. N. Abbasi and H. Mottaghi Golshan: Caristi's fixed point theorem and its equivalences in fuzzy metric spaces. Kybernetika 52 (2016), 6, 966-979.   CrossRef
  2. A. Amini-Harandi: Some generalizations of Caristi's fixed point theorem with applications to the fixed point theory of weakly contractive set-valued maps and the minimization problem. Nonlinear Anal. 72 (2010), 12, 4661-4665.   DOI:10.1016/
  3. D. W. Boyd and J. S. W. Wong: On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458-464.   DOI:10.1016/S0009-9260(69)80106-6
  4. J. Caristi: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215 (1976), 241-251.   CrossRef
  5. J. Caristi and W. A. Kirk: Geometric fixed point theory and inwardness conditions. In: The geometry of metric and linear spaces (Proc. Conf., Michigan State Univ., East Lansing, Mich., 1974), pp. 74-83. Lecture Notes in Math., Vol. 490. Springer, Berlin 1975.   CrossRef
  6. S.-S. Chang, Y. J. Cho and S.-M. Kang: Nonlinear Operator Theory in Probablistic Metric Spaces. Nova Publishers, 2001.   CrossRef
  7. 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
  8. 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
  9. V. Gregori, J.-J. Miñana 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
  10. 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
  11. O. Hadzic and E. Pap: Fixed Point Theory in Probabilistic Metric Spaces. Springer Science and Business Media, 2013.   CrossRef
  12. J. R. Jachymski: Caristi's fixed point theorem and selections of set-valued contractions. J. Math. Anal. Appl. 227 (1998), 1, 55-67.   DOI:10.1006/jmaa.1998.6074
  13. M. A. Khamsi: Remarks on Caristi's fixed point theorem. Nonlinear Anal. 71 (2009), 1-2, 227-231.   DOI:10.1016/
  14. M. A. Khamsi and W. A. Kirk: An Introduction to Metric Spaces and Fixed Point Theory. John Wiley and Sons, 2011.   CrossRef
  15. M. A. Khamsi and D. Misane: Compactness of convexity structures in metrics paces. Math. Japon. 41 (1995), 321-326.   CrossRef
  16. E. Klement and R. Mesiar: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms. Elsevier Science, 2005.   CrossRef
  17. E. P. Klement, R. Mesiar and E. Pap: A characterization of the ordering of continuous t-norms. Fuzzy Sets and Systems 86 (1997), 2, 189-195.   DOI:10.1016/0165-0114(95)00407-6
  18. E. P. Klement, R. Mesiar and E. Pap: Triangular norms. Trends in Logic--Studia Logica Library 8, Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  19. E. P. Klement, R. Mesiar and E. Pap: Triangular norms. position paper ii: general constructions and parameterized families. Fuzzy Sets and Systems 145 (2004), 3, 411-438.   DOI:10.1016/j.fss.2003.10.003
  20. E. P. Klement, R. Mesiar and E. Pap: Triangular norms. position paper iii: continuous t-norms. Fuzzy Sets and Systems 145 (2004), 3, 439-454.   DOI:10.1016/S0165-0114(03)00304-X
  21. A. Kolesárová: A note on archimedean triangular norms. BUSEFAL 80 (1999), 57-60.   CrossRef
  22. I. Kramosil and J. Michálek: Fuzzy metrics and statistical metric spaces. Kybernetika 11 (1075), 5, 336-344.   CrossRef
  23. K. Menger: Statistical metrics. Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 535-537.   DOI:10.1073/pnas.28.12.535
  24. J. C. Moore: Mathematical Methods for Economic Theory 1. Springer Science and Business Media, 1999.   CrossRef
  25. B. Schweizer and A. Sklar: Statistical metric spaces. Pacific J. Math. 10 (1960), 313-334.   DOI:10.2140/pjm.1960.10.313
  26. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York 1983.   CrossRef
  27. S. Sedghi, N. Shobkolaei and I. Altun: A new approach to Caristi's fixed point theorem on non-Archimedean fuzzy metric spaces. Iran. J. Fuzzy Syst. 12 (2015), 2, 137-143, 157.   CrossRef
  28. E. Zeidler: Nonlinear functional analysis and its applications I (Fixed-point theorems). Springer-Verlag, New York 1986.   DOI:10.1007/978-1-4612-4838-5
  29. E. Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung. Math. Ann. 65 (1907), 1, 107-128.   DOI:10.1007/BF01450054