# ON COMPATIBLE MAPPINGS OF TYPE (I) AND (II) IN INTUITIONISTIC FUZZY METRIC SPACES

Alaca, Cihangir;Altun, Ishak;Turkoglu, Duran

• Published : 2008.07.31
• 79 7

#### Abstract

In this paper, we give some new definitions of compatible mappings in intuitionistic fuzzy metric spaces and we prove a common fixed point theorem for four mappings under the condition of compatible mappings of type (I) and of type (II) in complete intuitionistic fuzzy metric spaces.

#### Keywords

intuitionistic fuzzy metric space;common fixed point;compatible mappings and compatible mappings of types $({\alpha})$$({\beta})$;(I) and (II)

#### References

1. S. Banach, Theorie les operations lineaires, Manograie Mathematyezne Warsaw Poland, 1932
2. Z. K. Deng, Fuzzy pseudo-metric spaces, J. Math. Anal. Appl. 86 (1982), 74-95 https://doi.org/10.1016/0022-247X(82)90255-4
3. M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979), 338-353 https://doi.org/10.1016/0022-247X(79)90189-6
4. J. X. Fang, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 46 (1992), 107-113 https://doi.org/10.1016/0165-0114(92)90271-5
5. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), 395-399 https://doi.org/10.1016/0165-0114(94)90162-7
6. M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 27 (1988), 385-389 https://doi.org/10.1016/0165-0114(88)90064-4
7. V. Gregori, S. Romaguera, and P. Veeramani, A note on intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals 28 (2006), 902-905 https://doi.org/10.1016/j.chaos.2005.08.113
8. G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly. 83 (1976), 261-263 https://doi.org/10.2307/2318216
9. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems 12 (1984), 215-229 https://doi.org/10.1016/0165-0114(84)90069-1
10. I. Kramosil and J. Michalek, Fuzzy metric and Statistical metric spaces, Kybernetica 11 (1975), 326-334
11. R. Lowen, Fuzzy Set Theory, Kluwer Academic Pub., Dordrecht 1996
12. R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl. 188 (1994), 436-440 https://doi.org/10.1006/jmaa.1994.1437
13. J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals 22 (2004), 1039-1046 https://doi.org/10.1016/j.chaos.2004.02.051
14. J. S. Park, Y. C. Kwun, and J. H. Park, A fixed point theorem in the intuitionistic fuzzy metric spaces, Far East J. Math. Sci. 16 (2005), 137-149
15. R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons & Fractals 27 (2006), 331-344 https://doi.org/10.1016/j.chaos.2005.03.019
16. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 314-334
17. D. Turkoglu, C. Alaca, Y. J. Cho, and C. Yildiz, Common fixed point theorems in intuitionistic fuzzy metric spaces, J. Appl. Math. & Computing 22 (2006), 411-424 https://doi.org/10.1007/BF02896489
18. D. Turkoglu, C. Alaca, and C. Yildiz, Compatible maps and compatible maps of types (${\alpha}$) and (${\beta}$) in intuitionistic fuzzy metric spaces, Demonstratio Math. 39 (2006), 671-684
19. D. Turkoglu, I. Altun, and Y. J. Cho, Common fixed points of compatible mappings of type (I) and (II) in fuzzy metric spaces, J. Fuzzy Math. 15 (2007), 435-448
20. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338-353 https://doi.org/10.1016/S0019-9958(65)90241-X
21. C. Alaca, D. Turkoglu, and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals 29 (2006), 1073-1078 https://doi.org/10.1016/j.chaos.2005.08.066
22. D. Mihet¸, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007), 915-921 https://doi.org/10.1016/j.fss.2006.11.012
23. M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79 https://doi.org/10.1112/jlms/s1-37.1.74

#### Cited by

1. Some new fixed point results on intuitionistic fuzzy metric spaces vol.3, pp.1, 2016, https://doi.org/10.1080/23311835.2016.1142839
2. Fixed Point Theorem for Compatible Maps with Type(I) and (II) in Intuitionistic Fuzzy Metric Space vol.10, pp.3, 2010, https://doi.org/10.5391/IJFIS.2010.10.3.194
3. Common fixed point theorems for families of maps in complete L-fuzzy metric spaces vol.23, pp.3, 2009, https://doi.org/10.2298/FIL0903067H
4. Common fixed point theorems for families of compatible mappings in intuitionistic fuzzy metric spaces vol.56, pp.2, 2010, https://doi.org/10.1007/s11565-010-0105-1
5. Common Fixed Point Theorems in Intuitionistic Fuzzy Metric Spaces vol.01, pp.06, 2010, https://doi.org/10.4236/am.2010.16067