Surveys in Mathematics and its Applications


ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 8 (2013), 1 -- 10

A COMMON FIXED POINT THEOREM IN
MENGER SPACE USING IMPLICIT RELATION

B.D. Pant and Sunny Chauhan

Abstract. The main purpose of this paper is to prove a common fixed point theorem for two pairs of weakly compatible mappings in Menger space using implicit relation.

2010 Mathematics Subject Classification: 54H25; 47H10.
Keywords: Triangle function (t-norm), Menger space; Implicit relation; Common fixed point; Compatible maps; Weakly compatible maps.

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References

  1. S.S. Chang, Y.J. Cho and S.M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Inc., New York, (2001). MR1911759. Zbl 1080.47054.

  2. S. Chauhan and B.D. Pant, Common fixed point theorems for occasionally weakly compatible mappings using implicit relation, Journal of the Indian Math. Soc., 77(1-4)(2010) 13-21.

  3. G. Jungck, Compatible mappings and common fixed points, Int. J. Math. & Math. Sci., 9(1986), 771-779. MR0870534. Zbl 0613.54029.

  4. G. Jungck and B.E. Rhoades, Fixed points for set valued functions without continuity, Indian J. Pure Appl. Math., 29(3)(1998), 227-238. MR1617919. Zbl 09047.54034.

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

  6. D. Miheţ, A generalization of a contraction principle in probabilistic metric spaces, Part II, Int. J. Math. & Math. Sci., 2005 (2005), 729-736. MR.2173690 Zbl 1083.54535.

  7. S.N. Mishra, Common fixed points of compatible mappings in PM-spaces, Math. Japon., 36 (1991), 283-289. MR1095742. Zbl 0731.54037.

  8. D. O'Regan, R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput., 195(2008), 86-94. MR2379198. Zbl 1135.54315.

  9. B.D. Pant and S. Chauhan, Common fixed point theorems for semicompatible mappings using implicit relation, Int. J. Math. Anal., 3(28)(2009), 1389-1398. MR2604831. Zbl 1196.54080.

  10. V. Popa, Fixed points for non-surjective expansion mappings satisfying an implicit relation, Bul. Ştiinţ. Univ. Baia Mare Ser. B Fasc. Mat.-Inform., 18(2002), 105-108. MR2014290. Zbl 1031.47034.

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

  12. B. Schweizer and A. Sklar, Probabilistic metric spaces, North-Holland Publishing Co., New York, (1983). MR790314. Zbl 0546.60010.

  13. V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory, 6(1972), 97-102. MR0310858 Zbl 0244.60004.

  14. S. Shakeri, A contraction theorem in Menger probabilistic metric spaces, J. Nonlinear Sci. Appl., 1(3) (2008), 189-193. MR2486960 Zbl 1160.54330.

  15. 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.

  16. B. Singh and S. Jain, Semicompatibility and fixed point theorems in fuzzy metric space using implicit relation, Int. J. Math. & Math. Sci., 2005(2005), 2617-2629. MR2184754. Zbl 1087.54506.



B.D. Pant Sunny Chauhan
Government Degree Collage, R.H. Government Postgraduate College,
Champawat, 262523, Uttarakhand, India. Kashipur, 244713, Uttarakhand, India.
E-mail: sun.gkv@gmail.com




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