Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property

Affiliation(s)

R. H. Government Postgraduate College, Kashipur, India.

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand.

R. H. Government Postgraduate College, Kashipur, India.

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand.

ABSTRACT

In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

In this paper, we prove a common fixed point theorem for a pair of weakly compatible mappings in fuzzy metric space using the joint common limit in the range property of mappings called (JCLR) property. An example is also furnished which demonstrates the validity of main result. We also extend our main result to two finite families of self mappings. Our results improve and generalize results of Cho et al. [Y. J. Cho, S. Sedghi and N. Shobe, “Generalized fixed point theorems for compatible mappings with some types in fuzzy metric spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244.] and several known results existing in the literature.

KEYWORDS

Fuzzy Metric Space; Weakly Compatible Mappings; (E.A) Property; (CLR) Property; (JCLR) Property

Fuzzy Metric Space; Weakly Compatible Mappings; (E.A) Property; (CLR) Property; (JCLR) Property

Cite this paper

S. Chauhan, W. Sintunavarat and P. Kumam, "Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property,"*Applied Mathematics*, Vol. 3 No. 9, 2012, pp. 976-982. doi: 10.4236/am.2012.39145.

S. Chauhan, W. Sintunavarat and P. Kumam, "Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces Using (JCLR) Property,"

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] I. Kramosil and J. Michalek, “Fuzzy Metric and Statistical Metric Spaces,” Kybernetika (Prague), Vol. 11, No. 5, 1975, pp. 336-344.

[3] A. George and P. Veeramani, “On Some Result in Fuzzy Metric Space,” Fuzzy Sets & Systems, Vol. 64, No. 3, 1994, 395-399. doi:10.1016/0165-0114(94)90162-7

[4] M. S. El Naschie, “On a Fuzzy Kahler-Like Manifold Which Is Consistent with Two-Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 95-98. doi:10.1515/IJNSNS.2005.6.2.95

[5] M. Aamri and D. El Moutawakil, “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

[6] I. Aalam, S. Kumar and B. D. Pant, “A Common Fixed Point Theorem in Fuzzy Metric Space,” Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 4, 2010, pp. 76-82.

[7] M. Abbas, I. Altun and D. Gopal, “Common 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.

[8] S. Kumar, “Fixed Point Theorems for Weakly Compatible Maps under E.A. Property in Fuzzy Metric Spaces,” Applied Mathematics & Information Sciences, Vol. 29, No. 1-2, 2011, pp. 395-405.

[9] S. Kumar and B. Fisher, “A Common Fixed Point Theorem in Fuzzy Metric Space Using Property (E.A.) and Implicit Relation,” Thai Journal of Mathematics, Vol. 8, No. 3, 2010, pp. 439-446.

[10] D. Mihet, “Fixed Point Theorems in Fuzzy Metric Spaces Using Property E.A.,” Nonlinear Analysis, Vol. 73, No. 7, 2010, pp. 2184-2188. doi:10.1016/j.na.2010.05.044

[11] S. Sedghi, C. Alaca and N. Shobe, “On Fixed Points of Weakly Commuting Mappings with Property (E.A),” Journal of Advanced Studies in Topology, Vol. 3, No. 3, 2012.

[12] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces,” Journal of Applied Mathematics, Vol. 2011, 2011, Article ID 637958.

[13] W. Sintunavarat and P. Kumam, “Common Fixed Points for R-Weakly Commuting in Fuzzy Metric Spaces,” Annali dell'Universita di Ferrara, 2012 (in press). doi:10.1007/s11565-012-0150-z

[14] Y. J. Cho, “Fixed Points in Fuzzy Metric Spaces,” Journal of Fuzzy Mathematics, Vol. 5, No. 4, 1997, pp. 949-962.

[15] M. Imdad and J. Ali, “Some Common Fixed Point Theorems in Fuzzy Metric Spaces,” Mathematical Communications, Vol. 11, No. 2, 2006, pp. 153-163.

[16] S. Kutukcu, D. Turkoglu and C. Yildiz, “Common Fixed Points of Compatible Maps of Type (β) on Fuzzy Metric Spaces,” Communications of the Korean Mathematical Society, Vol. 21, No. 1, 2006, pp. 89-100.

[17] P. P. Murthy, S. Kumar and K. Tas, “Common Fixed Points of Self Maps Satisfying an Integral Type Contractive Condition in Fuzzy Metric Spaces,” Mathematical Communications, Vol. 15, No. 2, 2010, pp. 521-537.

[18] D. O’Regan and M. Abbas, “Necessary and Sufficient Conditions for Common Fixed Point Theorems in Fuzzy Metric Space,” Demonstratio Mathematica, Vol. 42, No. 4, 2009, pp. 887-900.

[19] B. D. Pant and S. Chauhan, “Common Fixed Point Theorems for Two Pairs of Weakly Compatible Mappings in Menger Spaces and Fuzzy Metric Spaces,” Scientific Studies and Research. Series Mathematics and Informatics, Vol. 21, No. 2, 2011, pp. 81-96.

[20] S. Sedghi, D. Turkoglu and N. Shobe, “Common Fixed Point of Compatible Maps of Type (γ) on Complete Fuzzy Metric Spaces,” Communications of the Korean Mathematical Society, Vol. 24, No. 4, 2009, pp. 581-594. doi:10.4134/CKMS.2009.24.4.581

[21] S. L. Singh and A. Tomar, “Fixed Point Theorems in FM-Spaces,” Journal of Fuzzy Mathematics, Vol. 12, No. 4, 2004, pp. 845-859

[22] W. Sintunavarat, Y. J. Cho and P. Kumam, “Coupled Coincidence Point Theorems for Contractions without Commutative Condition in Intuitionistic Fuzzy Normed Spaces,” Fixed Point Theory and Applications, Vol. 2011, No. 81, 2011.

[23] W. Sintunavarat and P. Kumam, “Fixed Point Theorems for a Generalized Intuitionistic Fuzzy Contraction in Intuitionistic Fuzzy Metric Spaces,” Thai Journal of Mathematics, Vol. 10 No. 1, 2012, pp. 123-135.

[24] P. V. Subrahmanyam, “A Common Fixed Point Theorem in Fuzzy Metric Spaces,” Information Sciences, Vol. 83, No. 3-4, 1995, pp. 109-112. doi:10.1016/0020-0255(94)00043-B

[25] R. Vasuki, “Common Fixed Points for R-Weakly Commuting Maps in Fuzzy Metric Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 4, 1999, pp. 419-423.

[26] Y. J. Cho, S. Sedghi and N. Shobe, “Generalized Fixed Point Theorems for Compatible Mappings with Some Types in Fuzzy Metric Spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244. doi:10.1016/j.chaos.2007.06.108

[27] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” In North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983.

[28] M. Grabiec, “Fixed Points in Fuzzy Metric Spaces,” Fuzzy Sets & System, Vol. 27, No. 3, 1988, pp. 385-389. doi:10.1016/0165-0114(88)90064-4

[29] S. N. Mishra, N. Sharma and S. L. Singh, “Common Fixed Points of Maps on Fuzzy Metric Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 17, No. 2, 1994, pp. 253-258. doi:10.1155/S0161171294000372

[30] G. Jungck, “Common Fixed Points for Noncontinuous Nonself Maps on Nonmetric Spaces,” Far East Journal of Mathematical Sciences Vol. 4, No. 2, 1996, pp. 199215.

[31] H. K. Pathak, R. R. López and R. K. Verma, “A Common Fixed Point Theorem Using Implicit Relation and Property (E.A) in Metric Spaces,” Filomat, Vol. 21, No. 2, 2007, pp. 211-234. doi:10.2298/FIL0702211P

[32] M. Imdad, J. Ali and M. Tanveer, “Coincidence and Common Fixed Point Theorems for Nonlinear Contractions in Menger PM Spaces,” Chaos, Solitons & Fractals, Vol. 42, No. 5, 2009, pp. 3121-3129. doi:10.1016/j.chaos.2009.04.017

[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] I. Kramosil and J. Michalek, “Fuzzy Metric and Statistical Metric Spaces,” Kybernetika (Prague), Vol. 11, No. 5, 1975, pp. 336-344.

[3] A. George and P. Veeramani, “On Some Result in Fuzzy Metric Space,” Fuzzy Sets & Systems, Vol. 64, No. 3, 1994, 395-399. doi:10.1016/0165-0114(94)90162-7

[4] M. S. El Naschie, “On a Fuzzy Kahler-Like Manifold Which Is Consistent with Two-Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 95-98. doi:10.1515/IJNSNS.2005.6.2.95

[5] M. Aamri and D. El Moutawakil, “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

[6] I. Aalam, S. Kumar and B. D. Pant, “A Common Fixed Point Theorem in Fuzzy Metric Space,” Bulletin of Mathematical Analysis and Applications, Vol. 2, No. 4, 2010, pp. 76-82.

[7] M. Abbas, I. Altun and D. Gopal, “Common 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.

[8] S. Kumar, “Fixed Point Theorems for Weakly Compatible Maps under E.A. Property in Fuzzy Metric Spaces,” Applied Mathematics & Information Sciences, Vol. 29, No. 1-2, 2011, pp. 395-405.

[9] S. Kumar and B. Fisher, “A Common Fixed Point Theorem in Fuzzy Metric Space Using Property (E.A.) and Implicit Relation,” Thai Journal of Mathematics, Vol. 8, No. 3, 2010, pp. 439-446.

[10] D. Mihet, “Fixed Point Theorems in Fuzzy Metric Spaces Using Property E.A.,” Nonlinear Analysis, Vol. 73, No. 7, 2010, pp. 2184-2188. doi:10.1016/j.na.2010.05.044

[11] S. Sedghi, C. Alaca and N. Shobe, “On Fixed Points of Weakly Commuting Mappings with Property (E.A),” Journal of Advanced Studies in Topology, Vol. 3, No. 3, 2012.

[12] W. Sintunavarat and P. Kumam, “Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces,” Journal of Applied Mathematics, Vol. 2011, 2011, Article ID 637958.

[13] W. Sintunavarat and P. Kumam, “Common Fixed Points for R-Weakly Commuting in Fuzzy Metric Spaces,” Annali dell'Universita di Ferrara, 2012 (in press). doi:10.1007/s11565-012-0150-z

[14] Y. J. Cho, “Fixed Points in Fuzzy Metric Spaces,” Journal of Fuzzy Mathematics, Vol. 5, No. 4, 1997, pp. 949-962.

[15] M. Imdad and J. Ali, “Some Common Fixed Point Theorems in Fuzzy Metric Spaces,” Mathematical Communications, Vol. 11, No. 2, 2006, pp. 153-163.

[16] S. Kutukcu, D. Turkoglu and C. Yildiz, “Common Fixed Points of Compatible Maps of Type (β) on Fuzzy Metric Spaces,” Communications of the Korean Mathematical Society, Vol. 21, No. 1, 2006, pp. 89-100.

[17] P. P. Murthy, S. Kumar and K. Tas, “Common Fixed Points of Self Maps Satisfying an Integral Type Contractive Condition in Fuzzy Metric Spaces,” Mathematical Communications, Vol. 15, No. 2, 2010, pp. 521-537.

[18] D. O’Regan and M. Abbas, “Necessary and Sufficient Conditions for Common Fixed Point Theorems in Fuzzy Metric Space,” Demonstratio Mathematica, Vol. 42, No. 4, 2009, pp. 887-900.

[19] B. D. Pant and S. Chauhan, “Common Fixed Point Theorems for Two Pairs of Weakly Compatible Mappings in Menger Spaces and Fuzzy Metric Spaces,” Scientific Studies and Research. Series Mathematics and Informatics, Vol. 21, No. 2, 2011, pp. 81-96.

[20] S. Sedghi, D. Turkoglu and N. Shobe, “Common Fixed Point of Compatible Maps of Type (γ) on Complete Fuzzy Metric Spaces,” Communications of the Korean Mathematical Society, Vol. 24, No. 4, 2009, pp. 581-594. doi:10.4134/CKMS.2009.24.4.581

[21] S. L. Singh and A. Tomar, “Fixed Point Theorems in FM-Spaces,” Journal of Fuzzy Mathematics, Vol. 12, No. 4, 2004, pp. 845-859

[22] W. Sintunavarat, Y. J. Cho and P. Kumam, “Coupled Coincidence Point Theorems for Contractions without Commutative Condition in Intuitionistic Fuzzy Normed Spaces,” Fixed Point Theory and Applications, Vol. 2011, No. 81, 2011.

[23] W. Sintunavarat and P. Kumam, “Fixed Point Theorems for a Generalized Intuitionistic Fuzzy Contraction in Intuitionistic Fuzzy Metric Spaces,” Thai Journal of Mathematics, Vol. 10 No. 1, 2012, pp. 123-135.

[24] P. V. Subrahmanyam, “A Common Fixed Point Theorem in Fuzzy Metric Spaces,” Information Sciences, Vol. 83, No. 3-4, 1995, pp. 109-112. doi:10.1016/0020-0255(94)00043-B

[25] R. Vasuki, “Common Fixed Points for R-Weakly Commuting Maps in Fuzzy Metric Spaces,” Indian Journal of Pure and Applied Mathematics, Vol. 30, No. 4, 1999, pp. 419-423.

[26] Y. J. Cho, S. Sedghi and N. Shobe, “Generalized Fixed Point Theorems for Compatible Mappings with Some Types in Fuzzy Metric Spaces,” Chaos, Solitons & Fractals, Vol. 39, No. 5, 2009, pp. 2233-2244. doi:10.1016/j.chaos.2007.06.108

[27] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” In North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing Co., New York, 1983.

[28] M. Grabiec, “Fixed Points in Fuzzy Metric Spaces,” Fuzzy Sets & System, Vol. 27, No. 3, 1988, pp. 385-389. doi:10.1016/0165-0114(88)90064-4

[29] S. N. Mishra, N. Sharma and S. L. Singh, “Common Fixed Points of Maps on Fuzzy Metric Spaces,” International Journal of Mathematics and Mathematical Sciences, Vol. 17, No. 2, 1994, pp. 253-258. doi:10.1155/S0161171294000372

[30] G. Jungck, “Common Fixed Points for Noncontinuous Nonself Maps on Nonmetric Spaces,” Far East Journal of Mathematical Sciences Vol. 4, No. 2, 1996, pp. 199215.

[31] H. K. Pathak, R. R. López and R. K. Verma, “A Common Fixed Point Theorem Using Implicit Relation and Property (E.A) in Metric Spaces,” Filomat, Vol. 21, No. 2, 2007, pp. 211-234. doi:10.2298/FIL0702211P

[32] M. Imdad, J. Ali and M. Tanveer, “Coincidence and Common Fixed Point Theorems for Nonlinear Contractions in Menger PM Spaces,” Chaos, Solitons & Fractals, Vol. 42, No. 5, 2009, pp. 3121-3129. doi:10.1016/j.chaos.2009.04.017