Coupled Fixed Point Theorem for Weakly Compatible Mappings in Menger Spaces

Affiliation(s)

National Institute of Technology, Hamirpur, India.

Department of Mathematics, Ahir College, Rewari, India.

Department of Mathematics, DCRUST, Murthal, Sonepat, India.

National Institute of Technology, Hamirpur, India.

Department of Mathematics, Ahir College, Rewari, India.

Department of Mathematics, DCRUST, Murthal, Sonepat, India.

ABSTRACT

In this paper, first, we introduce the
notion of weakly compatible maps for coupled maps and then prove a coupled
fixed point theorem under more general *t*-norm(*H*-type norm) in
Menger spaces. We support our theorem by providing a suitable example. At
the end, we obtain an application.

Cite this paper

Grewal, M. , Jain, M. , Vats, R. and Kumar, S. (2013) Coupled Fixed Point Theorem for Weakly Compatible Mappings in Menger Spaces.*Applied Mathematics*, **4**, 1714-1719. doi: 10.4236/am.2013.412234.

Grewal, M. , Jain, M. , Vats, R. and Kumar, S. (2013) Coupled Fixed Point Theorem for Weakly Compatible Mappings in Menger Spaces.

References

[1] K. Menger, “Statistical Metrices,” Proceedings of the National Academy of Sciences of USA, Vol. 28, 1942, pp. 535-537. http://dx.doi.org/10.1073/pnas.28.12.535

[2] V. M. Sehgal and A. T. Bharucha-Reid, “Fixed Points of Contraction Mappings on Probabilistic Metric Spaces,” Mathematical Systems Theory, Vol. 6, No. 1-2, 1972, pp. 97-102.

http://dx.doi.org/10.1007/BF01706080

[3] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” North Holland Series in Probability and Applied Mathematics, Vol. 5, 1983.

[4] S. N. Mishra, “Common Fixed Points of Compatible Mappings in PM-Spaces,” Mathematica Japonica, Vol. 36, 1991, pp. 283-289.

[5] G. Jungck, “Common Fixed Points for Non-Continuous Non-Self Maps on Non-Metric Spaces,” Far East Journal of Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199215.

[6] B. Singh and S. Jain, “A Fixed Point Theorem in Menger Space through Weak Compatibility,” Journal of Mathematical Analysis and Applications, Vol. 301, 2005, pp. 439-448.

http://dx.doi.org/10.1016/j.jmaa.2004.07.036

[7] J. X. Fang, “Common Fixed Point Theorems of Compatible and Weakly Compatible Maps in Menger Spaces,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 71, No. 5-6, 2009, pp. 1833-1843.

[8] O. Hadzic and E. Pap, “Fixed Point Theory in Probabilistic Metric Spaces, Vol. 536 of Mathematics and Its Applications,” Kluwer Academic, Dordrecht, 2001.

[9] T. G. Bhaskar and V. Lakshmikantham, “Fixed Point Theorems in Partially Ordered Metric Spaces and Applications,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 65, No. 7, 2006, pp. 1379-1393.

http://dx.doi.org/10.1016/j.na.2005.10.017

[10] V. Lakshmikantham and L. Ciric, “Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 70, No. 12, 2009, pp. 4341-4349.

[11] M. Abbas, M. Ali Khan and S. Redenovic, “Common Coupled Fixed Point Theorems in Cone Metric Spaces for W-Compatible Mappings,” Applied Mathematics and Computation, Vol. 217, No. 1, 2010, pp. 195-202.

http://dx.doi.org/10.1016/j.amc.2010.05.042

[1] K. Menger, “Statistical Metrices,” Proceedings of the National Academy of Sciences of USA, Vol. 28, 1942, pp. 535-537. http://dx.doi.org/10.1073/pnas.28.12.535

[2] V. M. Sehgal and A. T. Bharucha-Reid, “Fixed Points of Contraction Mappings on Probabilistic Metric Spaces,” Mathematical Systems Theory, Vol. 6, No. 1-2, 1972, pp. 97-102.

http://dx.doi.org/10.1007/BF01706080

[3] B. Schweizer and A. Sklar, “Probabilistic Metric Spaces,” North Holland Series in Probability and Applied Mathematics, Vol. 5, 1983.

[4] S. N. Mishra, “Common Fixed Points of Compatible Mappings in PM-Spaces,” Mathematica Japonica, Vol. 36, 1991, pp. 283-289.

[5] G. Jungck, “Common Fixed Points for Non-Continuous Non-Self Maps on Non-Metric Spaces,” Far East Journal of Mathematical Sciences, Vol. 4, No. 2, 1996, pp. 199215.

[6] B. Singh and S. Jain, “A Fixed Point Theorem in Menger Space through Weak Compatibility,” Journal of Mathematical Analysis and Applications, Vol. 301, 2005, pp. 439-448.

http://dx.doi.org/10.1016/j.jmaa.2004.07.036

[7] J. X. Fang, “Common Fixed Point Theorems of Compatible and Weakly Compatible Maps in Menger Spaces,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 71, No. 5-6, 2009, pp. 1833-1843.

[8] O. Hadzic and E. Pap, “Fixed Point Theory in Probabilistic Metric Spaces, Vol. 536 of Mathematics and Its Applications,” Kluwer Academic, Dordrecht, 2001.

[9] T. G. Bhaskar and V. Lakshmikantham, “Fixed Point Theorems in Partially Ordered Metric Spaces and Applications,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 65, No. 7, 2006, pp. 1379-1393.

http://dx.doi.org/10.1016/j.na.2005.10.017

[10] V. Lakshmikantham and L. Ciric, “Coupled Fixed Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 70, No. 12, 2009, pp. 4341-4349.

[11] M. Abbas, M. Ali Khan and S. Redenovic, “Common Coupled Fixed Point Theorems in Cone Metric Spaces for W-Compatible Mappings,” Applied Mathematics and Computation, Vol. 217, No. 1, 2010, pp. 195-202.

http://dx.doi.org/10.1016/j.amc.2010.05.042