Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 8 (2013), 51 -- 57

COMMON FIXED POINT THEOREM FOR NONCOMPATIBLE MAPS
IN PROBABILISTIC METRIC SPACE

Suneel Kumar, Sunny Chauhan and B. D. Pant

Abstract. In this paper, we prove a common fixed point theorem for noncompatible maps in probabilistic metric space using implicit relation. Our result does not require either the completeness of the space or continuity of the maps.

2010 Mathematics Subject Classification: 54H25; 47H10.
Keywords: Probabilistic metric space; Noncompatible maps; Weakly compatible maps; Implicit relation; Property (E.A).

Full text

References

  1. M. Aamri and D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl., 270(2002), 181-188. MR1911759. Zbl 1008.54030.

  2. J.X. Fang and Y. Gao, Common fixed point theorems under strict contractive conditions in Menger spaces, Nonlinear Analysis,70(1)(2009), 184-193. MR2468228. Zbl 1170.47061.

  3. G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9(1986), 771-779. MR0870534. Zbl0613.54029.

  4. I. Kubiaczyk and S. Sharma, Some common fixed point theorems in Menger space under strict contractive conditions, Southeast Asian Bull. Math., 32(2008), 117-124. MR2385106. Zbl 05629600.

  5. S. Kumar and B.D. Pant, A common fixed point theorem in probabilistic metric space using implicit relation, Filomat, 22(2)(2008), 43-52. MR2484193. Zbl pre05619609.

  6. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. U.S.A., 28(1942), 535-537. MR0007576. Zbl 0063.03886.

  7. D. Mihet, A generalization of a contraction principle in probabilistic metric spaces, Part II, Int. J. Math. Math. Sci., 2005(5)(2005), 729-736. MR2173690. Zbl 1083.54535.

  8. S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon., 36 (1991), 283-289. MR1095742. Zbl0731.54037.

  9. R.P. Pant, Common fixed point theorems for contractive maps, J. Math. Anal. Appl., 226(1998), 251-258. MR1646430. Zbl0916.54027.

  10. B.D. Pant and S. Chauhan, Common fixed point theorems for semi compatible mappings using implicit relation, Int. J. Math. Anal., 5(28)(2009), 1389-1398. MR2604831.

  11. 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, 21(2)(2007), 211-234. MR2360891. Zbl 1141.54018.

  12. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math., 10(1960), 313-334. MR0115153. Zbl 0091.29801.

  13. B. Singh and S. Jain, A fixed point theorem in Menger Space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448. MR2105684. Zbl 1068.54044.




Suneel Kumar Sunny Chauhan
Government Higher Secondary School, Department of Mathematics,
Sanyasiowala, PO - Jaspur (U.S.Nagar), R.H. Government Postgraduate College,
Uttarakhand, India 244712. Kashipur, Uttarakhand, India 244713.
e-mail1: ksuneel_math@rediffmail.com e-mail: sun.gkv@gmail.com
e-mail2: suneelkchauhan@yahoo.in
B. D. Pant
Government Degree College,
Champawat, Uttarakhand, India.
e-mail: badridatt.pant@gmail.com


http://www.utgjiu.ro/math/sma