A New Approach to Caristi's Fixed Point Theorem on Non-Archimedean Fuzzy Metric Spaces

Document Type: Research Paper

Authors

1 Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran

2 Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran

3 Department of Mathematics, Faculty of Science and Arts, Kirikkale Uni- versity, 71450 Yahsihan, Kirikkale, Turkey

Abstract

In the present paper, we give a new approach to Caristi's fixed point
theorem on non-Archimedean fuzzy metric spaces. For this we define an
ordinary metric $d$ using the non-Archimedean fuzzy metric $M$ on a nonempty
set $X$ and we establish some relationship between $(X,d)$ and $(X,M,ast )$%
. Hence, we prove our result by considering the original Caristi's fixed
point theorem.

Keywords


[1] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung and S. M. Kang, Coincidence point and
minimization theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 88 (1997), 119-128.
[2] Y. J. Cho, Fixed points in fuzzy metric spaces, Journal of Fuzzy Mathematics, 5 (1997),
949-962.
[3] J. X. Fang, On xed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 46
(1992), 107-113.
[4] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems,
64 (1994), 395-399.
[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389.
[6] V. Gregori and A. Sapena, On xed-point theorem in fuzzy metric spaces, Fuzzy Sets and
Systems, 125 (2002), 245-252.
[7] O. Hadzic and E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Academic
Publishers, Dordrecht, 2001.
[8] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),
215-229.
[9] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11
(1975), 326-334.
[10] D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems,
158 (2007) 915-921.
[11] D. Mihet, Fuzzy  -contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets
and Systems, 159 (2008), 739-744.
[12] S. N. Mishra , S. N. Sharma and S. L. Singh, Common xed points of maps in fuzzy metric
spaces, Internat. J. Math. Math. Sci., 17 (1994), 253-258.

[13] V. Radu, Some remarks on the probabilistic contractions on fuzzy Menger spaces, In: The
Eighth International Conference on Appl. Math. Comput. Sci., Cluj-Napoca, 2002, Automat.
Comput. Appl. Math., 11 (2002), 125-131.
[14] B. Schweizer and A. Sklar, Statistical metric spaces, Paci c Journal of Mathematics, 10
(1960), 313-334.