Coincidence and Common Fixed Point of Weakly Compatible Maps in Fuzzy Metric Space

Affiliation(s)

School of Mathematics and Computer Applications, Thapar University, Patiala, India.

Department of Mathematics, Faculty of Science, Jazan University, Jazan, KSA.

School of Mathematics and Computer Applications, Thapar University, Patiala, India.

Department of Mathematics, Faculty of Science, Jazan University, Jazan, KSA.

ABSTRACT

The aim of this
paper is to establish some new common fixed point theorems for generalized contractive
maps in fuzzy metric space by using property (*E.A*.), common property (*E.A**.*), *JCLR*_{ST} property and *CLR*_{S} property. Our results
improve and extend the results of Chauhan *e**t* *al*. [1] and Sedghi *et* *al*. [2] besides several
known results. We also furnish an illustrative example in support of our
results.

KEYWORDS

*FM*-Space,
Weakly Compatible Maps,
*CLR*_{g} Property,
Property (*E.A*.),
The Common Property (*E.A.*),
*JCLR*_{ST} Property and Common Fixed Point

Cite this paper

Manro, S. and , S. (2014) Coincidence and Common Fixed Point of Weakly Compatible Maps in Fuzzy Metric Space.*Applied Mathematics*, **5**, 1335-1348. doi: 10.4236/am.2014.59126.

Manro, S. and , S. (2014) Coincidence and Common Fixed Point of Weakly Compatible Maps in Fuzzy Metric Space.

References

[1] Chauhan, S., Bhatnagar, S. and Radenovic, S. (2013) Common Fixed Point Theorem for Weakly Compatible Mappings in Fuzzy Metric Spaces. Le Matematiiche, 68, 87-98.

[2] Sedghi, S., Shobe, N. and Aliouche, A. (2010) A Common Fixed Point Theorem for Weakly Compatible Mappings in Fuzzy Metric Spaces. General Mathematics, 18, 3-12.

[3] Aamri, M. and El Moutawakil, D. (2002) Some New Common Fixed Point Theorems under Strict Contractive Conditions. Journal of Mathematical Analysis and Applications, 270, 181-188.

http://dx.doi.org/10.1016/S0022-247X(02)00059-8

[4] Chauhan, S., Sintunavarat, W. and Kumam, P. (2012) Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property. Applied Mathematics, 3, 976-982.

http://dx.doi.org/10.4236/am.2012.39145

[5] Fang, J.X. and Gao, Y. (2009) Common Fixed Point Theorems under Strict Contractive Conditions in Menger Spaces. Nonlinear Analysis, 70, 184-193.

http://dx.doi.org/10.1016/j.na.2007.11.045

[6] George, A. and Veeramani, P. (1994) On Some Results in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 64, 395-399.

http://dx.doi.org/10.1016/0165-0114(94)90162-7

[7] Grabiec, M. (1988) Fixed Points in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 27, 385-389.

http://dx.doi.org/10.1016/0165-0114(88)90064-4

[8] Gregori, V., Morillas, S. and Sapena, A. (2011) Examples of Fuzzy Metrics and Applications. Fuzzy Sets and Systems, 170, 95-111.

http://dx.doi.org/10.1016/j.fss.2010.10.019

[9] Kramosil, I. and Michalek, J. (1975) Fuzzy Metric and Statistical Spaces. Kybernetica, 11, 336-344.

[10] Kumar, S. and Fisher, B. (2010) A Common Fixed Point Theorem in Fuzzy Metric Space Using Property (E. A.) and Implicit Relation. Thai Journal of Mathematics, 8, 439-446.

[11] Mihet, D. (2010) Fixed Point Theorems in Fuzzy Metric Spaces Using Property E.A. Nonlinear Analysis, 73, 21842188.

http://dx.doi.org/10.1016/j.na.2010.05.044

[12] Shostak, A.P. (1989) Two Decades of Fuzzy Topology; Basic Ideas, Notions and Results. Russian Math Surveys, 44, 123-186.

http://dx.doi.org/10.1070/RM1989v044n06ABEH002295

[13] Sintunavarat, W. and Kumam, P. (2011) Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces. Journal of Applied Mathematics, 2011, Article ID: 637958.

[14] Liu, W., Wu, J. and Li, Z. (2005) Common Fixed Points of Single-Valued and Multivalued Maps. International Journal of Mathematics and Mathematical Sciences, 19, 3045-3055.

http://dx.doi.org/10.1155/IJMMS.2005.3045

[15] Manro, S., Kumar, S. and Bhatia, S.S. Common Fixed Point Theorems in Fuzzy Metric Spaces Using Common Range Property. Journal of Advanced Research in Applied Mathematics.

[16] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[17] Menger, K. (1942) Statistical Metrics. Proceedings of the National Academy of Sciences (USA), 28, 535-537.

http://dx.doi.org/10.1073/pnas.28.12.535

[18] Schweizer, B. and Sklar, A. (1983) Probabilistic Metric Spaces. Amsterdam.

[19] Jungck, G. (1986) Compatible Mappings and Common Fixed Points. International Journal of Mathematics and Mathematical Sciences, 9, 771-779.

http://dx.doi.org/10.1155/S0161171286000935

[20] Jungck, G. and Rhoades, B.E. (1998) Fixed Points for Set Valued Functions without Continuity. Indian Journal of Pure and Applied Mathematics, 29, 227-238.

[21] Imdad, M., Tanveer, M. and Hasan, M. (2010) Some Common Fixed Point Theorems in Menger PM-Spaces. Fixed Point Theory and Applications, 2010, Article ID: 819269.

[1] Chauhan, S., Bhatnagar, S. and Radenovic, S. (2013) Common Fixed Point Theorem for Weakly Compatible Mappings in Fuzzy Metric Spaces. Le Matematiiche, 68, 87-98.

[2] Sedghi, S., Shobe, N. and Aliouche, A. (2010) A Common Fixed Point Theorem for Weakly Compatible Mappings in Fuzzy Metric Spaces. General Mathematics, 18, 3-12.

[3] Aamri, M. and El Moutawakil, D. (2002) Some New Common Fixed Point Theorems under Strict Contractive Conditions. Journal of Mathematical Analysis and Applications, 270, 181-188.

http://dx.doi.org/10.1016/S0022-247X(02)00059-8

[4] Chauhan, S., Sintunavarat, W. and Kumam, P. (2012) Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property. Applied Mathematics, 3, 976-982.

http://dx.doi.org/10.4236/am.2012.39145

[5] Fang, J.X. and Gao, Y. (2009) Common Fixed Point Theorems under Strict Contractive Conditions in Menger Spaces. Nonlinear Analysis, 70, 184-193.

http://dx.doi.org/10.1016/j.na.2007.11.045

[6] George, A. and Veeramani, P. (1994) On Some Results in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 64, 395-399.

http://dx.doi.org/10.1016/0165-0114(94)90162-7

[7] Grabiec, M. (1988) Fixed Points in Fuzzy Metric Spaces. Fuzzy Sets and Systems, 27, 385-389.

http://dx.doi.org/10.1016/0165-0114(88)90064-4

[8] Gregori, V., Morillas, S. and Sapena, A. (2011) Examples of Fuzzy Metrics and Applications. Fuzzy Sets and Systems, 170, 95-111.

http://dx.doi.org/10.1016/j.fss.2010.10.019

[9] Kramosil, I. and Michalek, J. (1975) Fuzzy Metric and Statistical Spaces. Kybernetica, 11, 336-344.

[10] Kumar, S. and Fisher, B. (2010) A Common Fixed Point Theorem in Fuzzy Metric Space Using Property (E. A.) and Implicit Relation. Thai Journal of Mathematics, 8, 439-446.

[11] Mihet, D. (2010) Fixed Point Theorems in Fuzzy Metric Spaces Using Property E.A. Nonlinear Analysis, 73, 21842188.

http://dx.doi.org/10.1016/j.na.2010.05.044

[12] Shostak, A.P. (1989) Two Decades of Fuzzy Topology; Basic Ideas, Notions and Results. Russian Math Surveys, 44, 123-186.

http://dx.doi.org/10.1070/RM1989v044n06ABEH002295

[13] Sintunavarat, W. and Kumam, P. (2011) Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces. Journal of Applied Mathematics, 2011, Article ID: 637958.

[14] Liu, W., Wu, J. and Li, Z. (2005) Common Fixed Points of Single-Valued and Multivalued Maps. International Journal of Mathematics and Mathematical Sciences, 19, 3045-3055.

http://dx.doi.org/10.1155/IJMMS.2005.3045

[15] Manro, S., Kumar, S. and Bhatia, S.S. Common Fixed Point Theorems in Fuzzy Metric Spaces Using Common Range Property. Journal of Advanced Research in Applied Mathematics.

[16] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.

http://dx.doi.org/10.1016/S0019-9958(65)90241-X

[17] Menger, K. (1942) Statistical Metrics. Proceedings of the National Academy of Sciences (USA), 28, 535-537.

http://dx.doi.org/10.1073/pnas.28.12.535

[18] Schweizer, B. and Sklar, A. (1983) Probabilistic Metric Spaces. Amsterdam.

[19] Jungck, G. (1986) Compatible Mappings and Common Fixed Points. International Journal of Mathematics and Mathematical Sciences, 9, 771-779.

http://dx.doi.org/10.1155/S0161171286000935

[20] Jungck, G. and Rhoades, B.E. (1998) Fixed Points for Set Valued Functions without Continuity. Indian Journal of Pure and Applied Mathematics, 29, 227-238.

[21] Imdad, M., Tanveer, M. and Hasan, M. (2010) Some Common Fixed Point Theorems in Menger PM-Spaces. Fixed Point Theory and Applications, 2010, Article ID: 819269.