Common Fixed Points of Single and Multivalued Maps in Fuzzy Metric Spaces

Palaniappan Vijayaraju, Zahurullah Mohamed Ibrahim Sajath

.

DOI: 10.4236/am.2011.25079                

Abstract

In this paper we introduce the notion of common property (EA) in fuzzy metric spaces. Further we prove some common fixed points theorems for hybrid pair of single and multivalued maps under hybrid contractive conditions. Our results extend previous ones in fuzzy metric spaces.

Keywords

Fuzzy Metric Space, Common Fixed Point, Coincidence Point.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X
[2]	A. George and P. Veeramani, “On Some Results in Fuzzy Metric Spaces,” Fuzzy Sets and Systems, Vol. 64, No. 3, 1994, pp. 395-399. doi:10.1016/0165-0114(94)90162-7
[3]	I. Karmosil and J. Michalek, “Fuzzy Metric Statistical Metric Spaces,” Kybernetica, Vol. 11, No.5, 1975, pp. 336-344.
[4]	Z. K. Deng, “Fuzzy Pseudo-Metric Spaces,” Journal of Mathematical Analysis and Applications, Vol. 86, No. 1, 1982, pp. 74-95. doi:10.1016/0022-247X(82)90255-4
[5]	M. Grabiec, “Fixed Points in Fuzzy Metric Spaces,” Fuzzy Sets and Systems, Vol. 27, No. 3, 1988, pp. 385-389. doi:10.1016/0165-0114(88)90064-4
[6]	O. Kaleva and S. Seikkala, “On Fuzzy Metric Spaces,” Fuzzy Sets and Systems, Vol. 12, No. 3, 1984, pp. 215-229. doi:10.1016/0165-0114(84)90069-1
[7]	S. Sharma, “Common Fixed Point Theorems in Fuzzy Metric Spaces,” Fuzzy Sets and Systems, Vol. 127, No. 3, 2002, pp. 345-352. doi:10.1016/S0165-0114(01)00112-9
[8]	I. Kubiaczyk and S. Sharma, “Common Coincidence Point in Fuzzy Metric Spaces,” Journal of Fuzzy Mathematics, Vol. 11, No. 1, 2003, pp. 1-5.
[9]	B. Singh and M. S. Chauhan, “Common Fixed Points of Compatible Maps in Fuzzy Metric Spaces,” Fuzzy Sets and Systems, Vol. 115, No. 3, 2000, pp. 471-475. doi:10.1016/S0165-0114(98)00099-2
[10]	P. Vijayaraju and Z. M. I. Sajath, “Some Common Fixed Point Theorems in Fuzzy Metric Spaces,” International Journal of Mathematical Analysis, Vol. 3, No. 15, 2009, pp. 701-710.
[11]	M. Abbas, I. Altun and D. Gopal, “Comon Fixed Point Theorems for Non Compatible Mappings in Fuzzy Metric Spaces,” Bulletin of Mathematical Analysis and Applications, Vol. 1, No. 2, 2009, pp. 47-56.
[12]	G. Jungk, “Compatible Mappings and Common Fixed Points,” International Journal of Mathematical Analysis, Vol. 9, No. 4, 1986, pp. 771-779.
[13]	R. P. Pant, “Common Fixed Points of Non Commutative Mappings,” Journal of Mathematical Analysis and Applications, Vol. 188, No. 1, 1994, pp. 436-440. doi:10.1006/jmaa.1994.1437
[14]	A. Aamri and D. E. Moutawkil, “Some New Common Fixed Point Theorems under Strict Contractive Conditions,” Journal of Mathematical Analysis and Applications, Vol. 270, No. 1, 2002, pp. 181-188. doi:10.1016/S0022-247X(02)00059-8
[15]	Y. Liu, J. Wu and Z. Li, “Common Fixed Points of Single Valued and Multivalued Maps,” International Journal of Mathematics and Mathematical Sciences, Vol. 2005, No. 19, 2005, pp. 3045-3055. doi:10.1155/IJMMS.2005.3045
[16]	B. Schweizer and A. Sklar, “Statistical Metric Spaces,” Pacific Journal of Mathematics, Vol. 10, 1960, pp. 331- 334.
[17]	B. E. Rhoades and G. Jungck, “Fixed Points for Set Valued Function without Continuity,” Indian Journal of Pure and Applied Mathematics, Vol. 29, No. 3, 1998, pp. 227-238.
[18]	S. L. Singh and S. N. Mishra, “Coincidences and Fixed Points of Nonself Hybrid Contractions,” Journal of Mathematical Analysis and Applications, Vol. 256, No. 2, 2001, pp. 486-497. doi:10.1006/jmaa.2000.7301
[19]	T. Kamran, “Coincidence and Fixed Points for Hybrid Strict Contractions,” Journal of Mathematical Analysis and Applications, Vol. 299, No. 1, 2004, pp. 23-24.