Kybernetika 51 no. 2, 374-386, 2015

On best approximation in fuzzy metric spaces

Naser Abbasi and Hamid Mottaghi GolshanDOI: 10.14736/kyb-2015-2-0374


In this paper we introduce the notation of t-best approximatively compact sets, t-best approximation points, t-proximinal sets, t-boundedly compact sets and t-best proximity pair in fuzzy metric spaces. The results derived in this paper are more general than the corresponding results of metric spaces, fuzzy metric spaces, fuzzy normed spaces and probabilistic metric spaces.


best approximation, topology, fuzzy metric spaces


54A40, 41A50


  1. G. Beer and D. Pai: Proximal maps, prox maps and coincidence points. Numer. Funct. Anal. Optim. 11 (1990), 5-6, 429-448.   DOI:10.1080/01630569008816382
  2. S. S. Chang, Y. J. Cho and S. M. Kang: Nonlinear Operator Theory in Probabilistic Metric Spaces. Nova Science Publishers Inc, New York 2001.   CrossRef
  3. A. A. Eldred and P. Veeramani: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2006), 2, 1001-1006.   DOI:10.1016/j.jmaa.2005.10.081
  4. A. George and P. Veeramani: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64 (1994), 3, 395-399.   DOI:10.1016/0165-0114(94)90162-7
  5. A. George and P. Veeramani: On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems 90 (1997), 3, 365-368.   DOI:10.1016/s0165-0114(96)00207-2
  6. M. Grabiec: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 27 (1998), 3, 385-389.   DOI:10.1016/0165-0114(88)90064-4
  7. V. Gregori and S. Romaguera: Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems 144 (2004), 3, 411-420.   DOI:10.1016/s0165-0114(03)00161-1
  8. V. Gregori and S. Romaguera: Some properties of fuzzy metric spaces. Fuzzy Sets and Systems 115 (2000), 3, 485-489.   DOI:10.1016/s0165-0114(98)00281-4
  9. O. Hadžić and E. Pap: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic Publishers, Dordrecht 2001.   DOI:10.1007/978-94-017-1560-7
  10. P. C. Kainen: Replacing points by compacta in neural network approximation. J. Franklin Inst. 341 (2004), 4, 391-399.   DOI:10.1016/j.jfranklin.2004.03.001
  11. W. A. Kirk, P. S. Srinivasan and P. Veeramani: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4 (2003), 1, 79-89.   CrossRef
  12. I. Kramosil and J. Michálek: Fuzzy metrics and statistical metric spaces. Kybernetika 11 (1975), 5, 336-344.   CrossRef
  13. K. Menger: Statistical metrics. Proc. Nat. Acad. Sci. U. S., A. 28 (1942), 535-537.   DOI:10.1073/pnas.28.12.535
  14. J. Rodríguez-López and S. Romaguera: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 147 (2004), 2, 273-283.   DOI:10.1016/j.fss.2003.09.007
  15. S. Romaguera and M. Sanchis: On fuzzy metric groups. Fuzzy Sets and Systems 124 (2001), 1, 109-115.   DOI:10.1016/s0165-0114(00)00085-3
  16. B. E. Sahney and S. P. Singh: On best simultaneous approximation. In: Approximation Theory III (E. W. Cheney, ed.), 1980, pp. 783-789.   CrossRef
  17. B. Schweizer and A. Sklar: Statistical metric spaces. Pacific J. Math. 10 (1960), 314-334.   DOI:10.2140/pjm.1960.10.313
  18. M. Shams and S. M. Vaezpour: Best approximation on probabilistic normed spaces. Chaos Solitons Fractals 41 (2009), 4, 1661-1667.   DOI:10.1016/j.chaos.2008.07.009
  19. I. Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer-Verlag 1970.   DOI:10.1007/978-3-662-41583-2
  20. S. M. Vaezpour and F. Karimi: t-best approximation in fuzzy normed spaces. Iran. J. Fuzzy Syst. 5 (2008), 2, 93-99.   CrossRef
  21. P. Veeramani: Best approximation in fuzzy metric spaces. J. Fuzzy Math. 9 (2001), 1, 75-80.   CrossRef
  22. X. Xu: A result on best proximity pair of two sets. J. Approx. Theory 54 (1988), 3, 322-325.   DOI:10.1016/0021-9045(88)90008-1