DOI QR코드

DOI QR Code

FIXED POINT THEOREMS FOR WEAK CONTRACTION IN INTUITIONISTIC FUZZY METRIC SPACE

  • Vats, Ramesh Kumar (Department of Mathematics, National Institute of Technology) ;
  • Grewal, Manju (Department of Mathematics, National Institute of Technology)
  • Received : 2015.10.22
  • Accepted : 2016.04.11
  • Published : 2016.06.25

Abstract

The notion of weak contraction in intuitionistic fuzzy metric space is well known and its study is well entrenched in the literature. This paper introduces the notion of (${\psi},{\alpha},{\beta}$)-weak contraction in intuitionistic fuzzy metric space. In this contrast, we prove certain coincidence point results in partially ordered intuitionistic fuzzy metric spaces for functions which satisfy a certain inequality involving three control functions. In the course of investigation, we found that by imposing some additional conditions on the mappings, coincidence point turns out to be a fixed point. Moreover, we establish a theorem as an application of our results.

Keywords

common fixed point;fuzzy metric space;control function;weak contraction

References

  1. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
  2. C. Alaca, D. Turkoglu and C. Yildiz, Fixed points in intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 29 (2006), 1073-1078. https://doi.org/10.1016/j.chaos.2005.08.066
  3. I. Beg, C. Vetro, D. Gopal and M. Imdad, ($\phi$, $\psi$ )-weak contractions in intuitionistic fuzzy metric spaces, J. Intell. Fuzzy Systems, 26(5) (2014), 2497-2504.
  4. S. Banach, Theoriles operations Linearies Manograie Mathematyezne, Warsaw, Poland, 1932.
  5. M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74-79.
  6. M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems, 27 (1988), 385-389. https://doi.org/10.1016/0165-0114(88)90064-4
  7. G. Jungck and B.E. Rhoades, Fixed Point for set valued function without continuity, J. Pure Appl. Math., 29(3) (1998), 227-238.
  8. I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975), 336-344.
  9. M.S. Khan, M. Swaleh and S. Sessa, Fixed points theorems by altering distances between the points, Bull. Aust. Math.Soc., 30 (1984), 1-9. https://doi.org/10.1017/S0004972700001659
  10. A. Kumar and R. K. Vats, Common fixed point theorem in fuzzy metric space using control function, Commun. Korean Math. Soc., 28(3) (2013) 517-526. https://doi.org/10.4134/CKMS.2013.28.3.517
  11. R. Lowen, Fuzzy set theory, Kluwer Academic Publishers, Dordrecht, 1996.
  12. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA, 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535
  13. J.H. Park, Intutionistic fuzzy metric space, Chaos Solitons Fractals, 22 (2004), 1039-1046. https://doi.org/10.1016/j.chaos.2004.02.051
  14. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific. J. Math, 10 (1960), 313-334. https://doi.org/10.2140/pjm.1960.10.313
  15. D. Turkoglu, C. Alaca and C. Yildiz, Compatible maps and compatible maps of types ($\alpha$) and ($\beta$) in intuitionistic fuzzy metric spaces, Demonstratio Math., 39(3) (2006), 671-684.
  16. L.A. Zadeh, Fuzzy Sets, Inform. Control, 89 (1965), 338-353.
  17. http://en.wikipedia.org/wiki/Partially_ordered_set.