Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 12 (2017), 23 -- 34

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

SOME FIXED POINT RESULTS IN FUZZY METRIC SPACES USING A CONTROL FUNCTION

C. T. Aage, Binayak S. Choudhury and Krishnapada Das

Abstract. In this paper, we establish the results on existence and uniqueness of fixed point for φ-contractive and generalized C-contractive mapping in the fuzzy metric space in the sense of George and Veeramani. We use the notion of altering distance for proving the results.

2010 Mathematics Subject Classification: 47H10, 54H25.
Keywords: Probabilistic metric space; fuzzy metric space; altering distance; p-convergent subsequence.

Full text

References

  1. B.S. Choudhury and P.N. Dutta, A unified fixed point result in metric spaces involving a two variable function, FILOMAT, 14 (2000), 43-48. MR1953993.
    Zbl 1012.54047.

  2. B.S. Choudhury, A unique common fixed point theorem for a sequence of self-mappings in Menger spaces, Bull. Korean Math. Soc., 37(2000), No.3, 569-573. MR1779246. Zbl 0959.54026.

  3. B.S. Choudhury, A common unique fixed point result in metric spaces involv-ing generalised altering distances, Math. Communications, 10(2005), 105-110. MR2199098. Zbl 1089.54514.

  4. B.S. Choudhury and P.N. Dutta, Common fixed points for fuzzy mappings using generalised altering distances, Soochow J. Math., 31(2005), 71-81. MR2130497. Zbl 1068.54043

  5. B.S. Choudhury and K. Das, A new contraction principle in Menger spaces, Acta Mathematica Sinica, English Series, 24(8) (2008), 1379-1386. MR2438308.

  6. B.S. Choudhury, P.N. Dutta and K. Das, A fixed points result in Menger space using a real function, Acta. Math. Hungar., 122 (2009), 203-216. MR2480861.

  7. B.S. Choudhury and K. Das, A coincidence point result in Menger spaces using a control function, Chaos, Solitons and Fractals, 42(2009), 3058-3063. MR2560014. Zbl 1198.54072

  8. P.N. Dutta and B.S. Choudhury, A generalization of contraction mapping principle, Fixed Point Theory and Applications, Volume 2008, Article ID 406368, doi: 10.1155/2008/406368.

  9. P.N. Dutta, Binayak S. Choudhury and Krishnapada Das, Some Fixed Point Results in Menger Spaces Using a Control Function, Surveys in Mathematics and its Applications, 4 (2009), 41-52.MR2485791. Zbl 1180.54054.

  10. R. Farnoosh, A. Aghajani and P. Azhdari, Contraction theorems in fuzzy metric space, Chaos, Solitons and Fractals, 41 (2009), 854-858.

  11. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Systems, 64 (1994), 395-399.

  12. M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Systems, 27 (1988), 385-389.

  13. V. Gregori and S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets Systems, 144(3) (2004), 411-420. MR2061403.

  14. O. Hadzic and E. Pap, Fixed point theory in PM spaces, Dordrecht: Kluwer Academic Publishers; 2001.

  15. O. Hadzic and E. Pap, A fixed point theorem for multivalued mapping in probabilistic metric space and an application in fuzzy metric spaces, Fuzzy Sets Systems, 127 (2002). 333-344. MR1899066. Zbl 0994.47077

  16. T.L. Hicks, Fixed point theory in probabilistic metric spaces, Zb. Rad. Prirod. Mat. Fak. Ser. Mat., 13 (1983), 63-72.

  17. M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30(1984), 1-9.

  18. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric space, Kybernetika, 11 (1975), 336-344.

  19. K. Menger, Statistical metric, Proc. Nat. Acad. Sci. USA, 28 (1942), 535-537.

  20. D. Mihet, Multivalued generalisations of probabilistic contractions, J. Math. Anal. Appl., 304(2005), 464-472. MR2126543.

  21. D. Mihet, On fuzzy contractive mapping in fuzzy metric, Fuzzy Sets Systems, 158 (2007) 915-921. MR3226659.

  22. D. Mihet, Altering distances in probabilistic Menger spaces, Nonlinear Analysis: Theory, Methods & Applications, 71(2009), 2734-2738. MR2532798

  23. S.V.R. Naidu, Some fixed point theorems in metric spaces by altering distances, Czechoslovak Mathematical Journal, 53 (2003), 205-212. MR1962009.

  24. K.P.R. Sastry and G.V.R. Babu, Some fixed point theorems by altering distances between the points, Ind. J. Pure. Appl. Math., 30(6) (1999), 641-647. MR1701042.

  25. B. Schiweizer and A. Sklar, Probabilistic Metric Spaces, Amesterdam: North-Holland; 1983.

  26. V.M. Sehgal and A.T. Bharucha-Ried, Fixed points of contraction mapping on PM-spaces, Math. Syst. Theo., 6 (1972), 97-110.

  27. B. Singh and S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-48. MR2517633.

  28. R. Vasuki and P. Veeramani, Fixed point theorems and Cauchy sequences in fuzzy metric spaces, Fuzzy Sets and Systems, 135(2003), 415-417. MR1979610. Zbl 1029.54012.

  29. Tatjana Zikic-Dosenovic, A multivalued generalization of Hick's C-contraction, Fuzzy Sets and Systems, 151(2005), 549-562. MR2126173. Zbl 0574.54044.




C. T. Aage
School of Mathematical Sciences,
North Maharashtra University, Jalgaon, India.
e-mail: caage17@gmail.com


Binayak S. Choudhury
Department of Mathematics,
Bengal Engineering and Science University, Shibpur,
P.O. - B. Garden, Shibpur,
Howrah - 711103, West Bengal, India.
e-mail:binayak12@yahoo.co.in


Krishnapada Das
Department of Mathematics,
Bengal Engineering and Science University, Shibpur,
P.O. - B. Garden, Shibpur,
Howrah - 711103, West Bengal, India.
e-mail: kestapm@yahoo.co.in

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