Some Fixed Point Theorems of Semi Compatible and Occasionally Weakly Compatible Mappings in Menger Space

Y. Rohen Singh^{1, 2,} and L. Premila Devi²

¹NIT Manipur, Takyelpat, Imphal, Pin, Manipur, India

²D. M. College of Science, Imphal, Pin, Manipur, India

Cite this paper

Y. Rohen Singh and L. Premila Devi. Some Fixed Point Theorems of Semi Compatible and Occasionally Weakly Compatible Mappings in Menger Space. American Journal of Applied Mathematics and Statistics. 2015; 3(1):29-33. doi: 10.12691/AJAMS-3-1-6

Abstract

The notion of semi compatible mappings was introduced by Cho. Sharma and Sahu (Semicompatibility and fixed points, Math. Japon, 42(1), 1995, 91-98) and the notion of occationally weakly compatible mappings was introduced by Al-Thagafi M. A., Shahzad N. (Generalized I-non expansive selfmaps and invariant approximations, Acta. Math. Sinica (English series) 24(5), 2008, 867-876). In this paper, we prove a common fixed point theorem in Menger space using the concept of semi compatible and occasionally weakly compatible mappings. Some results are also given as corollaries. Our results generalise some similar results in the literature.

Keywords

common fixed point, compatible maps, menger space, probabilistic metric space, semi-compatible, weakly compatible, occasionally weakly compatible

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	K. Menger, Statistical Metric, Proc. Nat. Acad. Sci. 28 (1942), 535-537.
[2]	Sehgal V. M. and Bharucha-Reid A.T., Fixed points of contraction maps on probabilistic metric spaces, Math. System Theory 6(1972), 97-102.
[3]	S. Sessa, On a weak commutative condition in fixed point consideration, Publ. Inst. Math. (Beograd) 32 (1982) 149-153.
[4]	Schweizer B., Sklar A., Probanilistic Metric Spaces, Elsevier, North- Holland, New York (1983).
[5]	G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci. (1986) 771-779.
[6]	S. N. Mishra, Common fixed points of compatible mappings in PM- spaces, Math. Japon. 36(1991) 283-289.
[7]	Cho. Y. J., Sharma B. K. and Sahu R. D., Semi-compatibility and fixed points, Math. Japon. 42(1), 1995, 91-98.
[8]	Jungck G., Rhoades B.E., Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(1998), 227-238.
[9]	Singh B., Jain S., Semi-compatibility and fixed point theorems in Menger space using implicit relation, East Asian Math. J. 21(1), 2005, 65-76.
[10]	Singh B., Jain S., A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448.
[11]	Jungck G., Rhoades B.E., Fixed point theorems for occasionally weakly compatible mappings, Fixed point theory 7, 2(2006) 286-296.
[12]	Servet Kutukcu, A fixed point theorem in Menger spaces, International Mathematical Forum, 1(32), 2006, 1543-1554.
[13]	Al-Thagafi M. A., Shahzad N., Generalized I-non expansive selfmaps and invariant approximations, Acta. Math. Sinica (English series) 24, 5(2008) 867-876.
[14]	Chauhan S., Kumar S., Pant B. D., Common fixed point theorems for occasionally weakly compatible mappings in Menger spaces, J. Adv. Research Pure Math. 3, 4(2011) 17-23.
[15]	Arihant Jain and Basant Chaudhary, On common fixed point theorems for semi-compatible and occasionally weakly compatible mappings in Menger space, IJRAS, 14(3), 2013, 662-670.
[16]	Yumnam Rohen and Th. Chhatrajit, Semicompatible mappings and fixed point theorems in cone metris space, Wulfenia Journal, 21(4), 2014, 216-224.