# COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING CONTROL FUNCTION

• Kumar, Amit ;
• Vats, Ramesh Kumar
• Published : 2013.07.31
• 34 6

#### Abstract

We give a fixed point theorem for complete fuzzy metric space which generalizes fuzzy Banach contraction theorems established by V. Gregori and A. Spena [Fuzzy Sets and Systems 125 (2002), 245-252] using notion of altering distance, initiated by Khan et al. [Bull. Austral. Math. Soc. 30 (1984), 1-9] in metric spaces.

#### Keywords

common fixed point;fuzzy contractive mapping;complete fuzzy metric space

#### References

1. C. T. Aage and B. S. Choudhury, Some fixed point results in fuzzy metric spaces using a control function, to appear.
2. C. T. Aage and J. N. Salunke, On fixed point theorems in fuzzy metric spaces using a control function, Int. J. Nonlinear Anal. Appl. 2 (2011), no. 1, 50-57.
3. B. S. Choudhury and K. Das, A new contraction principle in Menger spaces, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 8, 1379-1386. https://doi.org/10.1007/s10114-007-6509-x
4. B. S. Choudhury, P. N. Dutta, and K. Das, A fixed points result in Menger space using a real function, Acta Math. Hungar. 122 (2009), no. 3, 203-216. https://doi.org/10.1007/s10474-008-7242-3
5. P. N. Dutta and B. S. Choudhury, A generalisation of contraction principle in met-ric spaces, Fixed Point Theory and Applications 2008 (2008), Article ID 406368, doi: 10.1155/2008/406368. https://doi.org/10.1155/2008/406368
6. P. N. Dutta, B. S. Choudhury, and Krishnapada Das, Some fixed point results in Menger spaces using a control function, Surv. Math. Appl. 4 (2009), 41-52.
7. A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems 90 (1997), no. 3, 365-368. https://doi.org/10.1016/S0165-0114(96)00207-2
8. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994), no. 3, 395-399. https://doi.org/10.1016/0165-0114(94)90162-7
9. V. Gregori and S. Romaguerab, Some properties of fuzzy metric spaces, Fuzzy Sets and Systems 115 (2000), no. 3, 485-489. https://doi.org/10.1016/S0165-0114(98)00281-4
10. V. Gregori and A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems 125 (2002), no. 2, 245-252. https://doi.org/10.1016/S0165-0114(00)00088-9
11. M. S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc. 30 (1984), no. 1, 1-9. https://doi.org/10.1017/S0004972700001659
12. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika (Prague) 11 (1975), no. 5, 326-334.
13. D. Mihet, Multivalued generalizations of probabilistic contractions, J. Math. Anal. Appl. 304 (2005), no. 2, 464-472. https://doi.org/10.1016/j.jmaa.2004.09.034
14. D. Mihet, On fuzzy contractive mapping in fuzzy metric, Fuzzy Sets and Systems 158 (2007), no. 8, 915-921. https://doi.org/10.1016/j.fss.2006.11.012
15. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 314-334.
16. R. Vasuki and P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems 135 (2003), no. 3, 415-417. https://doi.org/10.1016/S0165-0114(02)00132-X

#### Cited by

1. FIXED POINT THEOREMS FOR WEAK CONTRACTION IN INTUITIONISTIC FUZZY METRIC SPACE vol.38, pp.2, 2016, https://doi.org/10.5831/HMJ.2016.38.2.337