Caristi type fixed point theorems in fuzzy metric spaces

Abstract

In this paper, we extend the generalized Caristi's fixed point theorem proved by Bollenbacher and Hicks to $p$-orbitally complete fuzzy metric spaces by considering the fuzzy metric spaces in the sense of George and Veeramani. We also give some illustrative examples that support our results.

Keywords

• Caristi's fixed point theorem,fixed point,fuzzy metric space,$p$-complete fuzzy metric space

References

1. T. Abdeljawad and E. Karapınar, Quasicone metric spaces and generalizations of Caristi Kirk’s theorem, Fixed Point Theory Appl. Vol. 2009, ID 574387, 9 p, 2009.
2. Ö. Acar and I. Altun, Some generalizations of Caristi type fixed point metric spaces, Filomat 26(4), 833-837, 2012.
3. Ö. Acar, I. Altun and S. Romaguera, Caristi’s type mappings on complete partial metric spaces, Fixed Point Theory 14(1), 3-10, 2013.
4. R.P. Agarwal and M.A. Khamsi, Extension of Caristi’s fixed point theorem to vector valued metric spaces, Nonlinear Anal. 74, 141-145, 2011.
5. I. Altun and D. Mihet, Ordered non-archimedean fuzzy metric spaces and some fixed point results, Fixed Point Theory Appl. Vol. 2010, ID 782680, 11 p., 2010.
6. J.S. Bae, Fixed point theorems for weakly contractive multivalued maps, J. Math. Anal. Appl. 284, 690-697, 2003.
7. A. Bollenbacher and T.L. Hicks, A fixed point theorem revisited, Proc. Amer. Math. Soc. 102, 898-900, 1988.
8. J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215, 241-251, 1976.

9. L. Ciric, Periodic and fixed point theorems in a quasi-metric space, J. Austral. Math. Soc. Ser. A 54, 80-85, 1993.
10. Z. Deng, Fuzzy pseudometric spaces, J. Math. Anal. Appl. 86, 74-95, 1982.
11. C. Di Bari and C. Vetro, A fixed point theorem for a family of mappings in a fuzzy metric space, Rend. Circ. Mat. Palermo 52, 315-321, 2003.
12. C. Di Bari and C. Vetro, Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space, J. Fuzzy Math. 13, 973-982, 2005.
13. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47, 324-353, 1974.
14. M.A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69, 205-230, 1979.
15. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64, 395-399, 1994.
16. A. George and P. Veeramani, Some theorems in fuzzy metric spaces, J. Fuzzy Math. 3, 933-940, 1995.
17. A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90, 365-368, 1997.
18. M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27, 385-389, 1988.
19. V. Gregori, A. López-Crevillén, S. Morillas, and A. Sapena, On convergence in fuzzy metric spaces, Topology Appl. 156, 3002-3006, 2009.
20. V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115, 485-489, 2000.
21. V. Gregori and S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets and Systems 144, 411-420, 2004.
22. V. Gregori and A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125, 245-252, 2002.
23. A.M. Harder and L.M. Saliga, Periodic and fixed point theorems in $d$-complete topological spaces, Indian J. Pure Appl. Math. 26, 787-796, 1995.
24. T.L. Hicks, Fixed point theorems for quasi-metric spaces, Math. Japon. 33, 231-236, 1988.
25. T.L. Hicks, Fixed point theorems for $d$-complete topological spaces I, Internat. J. Math. Math. Sci. 15, 435-440, 1992.
26. C. Ionescu, Sh. Rezapour and M.E. Samei, Fixed points of a class of contractive-type multifunctions on fuzzy metric spaces, U.P.B. Sci. Bull., Series A 76, 3-12, 2014.
27. J.R. Jachymski, Caristi’s fixed point theorem and selections of set-valued contractions, J. Math. Anal. Appl. 227, 55-67, 1998.
28. O. Kada, T. Suzuki and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon. 44, 381-391, 1996.
29. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12, 215- 229, 1984.
30. E. Karapınar, Generalizations of Caristi Kirk’s theorem on partial metric spaces, Fixed Point Theory Appl. 2011:4, 2011.
31. E. Karapınar, I.M. Erhan and A. Öztürk, Fixed point theorems on quasi-partial metric spaces, Math. Com. Mod. 57, 2442-2448, 2013.
32. H. Karayılan and M. Telci, Common fixed point theorems for contractive type mappings in a fuzzy metric spaces, Rend. Circ. Mat. Palermo 60, 145-152, 2011.
33. M.A. Khamsi, Remarks on Caristi’s fixed point theorem, Nonlinear Anal. 71, 227-231, 2009.
34. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica 11, 336-344, 1975.
35. D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems 144, 431-439, 2004.
36. D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158, 915-921, 2007.
37. J. Rodríguez-López and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets and Systems 147, 273-283, 2004.
38. S. Romaguera, A Kirk type characterization of completeness for partial metric spaces, Fixed Point Theory Appl. Vol. 2010, ID 493298, 6 p, 2010.
39. I. Schweizer and A. Sklar, Statistical metric spaces, Pac. J. Math. 10, 314-334, 1960.
40. T. Suzuki, Generalized Caristi’s fixed point theorems by Bae and others, J. Math. Anal. Appl 302, 502-508, 2005.
41. R. Vasuki, A common fixed point theorem in a fuzzy metric space, Fuzzy Sets and Systems 97, 395-397, 1998.
42. C. Vetro and P. Vetro, Common fixed points for discontinuous mappings in fuzzy metric spaces, Rend. Circ. Mat. Palermo 57, 295-303, 2008.