Kybernetika 50 no. 5, 758-773, 2014

On the hyperspace of bounded closed sets under a generalized Hausdorff stationary fuzzy metric

Dong Qiu, Chongxia Lu, Shuai Deng and Liang WangDOI: 10.14736/kyb-2014-5-0758

Abstract:

In this paper, we generalize the classical Hausdorff metric with t-norms and obtain its basic properties. Furthermore, for a given stationary fuzzy metric space with a t-norm without zero divisors, we propose a method for constructing a generalized Hausdorff fuzzy metric on the set of the nonempty bounded closed subsets. Finally we discuss several important properties as completeness, completion and precompactness.

Keywords:

triangular norms, Hausdorff metric, hyperspace, stationary fuzzy metric

Classification:

46S40, 03E72, 54A40

paper.pdf

References:

  1. H. Adibi, Y. J. Cho, D. O'Regan and R. Saadati: Common fixed point theorems in $L$-fuzzy metric spaces. Appl. Math. Comput. 182 (2006), 820-828.   CrossRef
  2. J. P. Aubin and I. Ekeland: Applied Nonlinear Analysis. Wiley, New York 1984.   CrossRef
  3. D. Azé, J. N. Corvellec and R. E. Lucchetti: Variational pairs and applications to stability in nonsmooth analysis. Nonlinear Anal. Theory Methods Appl. 49 (2002), 643-670.   CrossRef
  4. G. Beer: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers, Dordrecht 1993.   CrossRef
  5. S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung and S. M. Kang: Coincidence point and minimization theorems in fuzzy metric spaces. Fuzzy Sets and Systems 88 (1997), 119-128.   CrossRef
  6. Y. J. Cho and N. Petrot: Existence theorems for fixed fuzzy points with closed $\alpha$-cut sets in complete metric spaces. Fuzzy Sets and Systems 26 (2011), 115-124.   CrossRef
  7. Z. K. Deng: Fuzzy pseudo-metric spaces. J. Math. Anal. Appl. 86 (1982), 74-95.   CrossRef
  8. R. Engelking: General Topology. PWN-Polish Science Publishers, Warsaw 1977.   CrossRef
  9. A. George and P. Veeramani: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 64 (1994), 395-399.   CrossRef
  10. B. M. Ghil and Y. K. Kim: Continuity of functions defined on the space of fuzzy sets. Inform. Sci. 157 (2003), 155-165.   CrossRef
  11. V. Gregori and A. Sapena: On fixed point theorem in fuzzy metric spaces. Fuzzy Sets and Systems 125 (2002), 245-252.   CrossRef
  12. V. Gregori and S. Romaguera: On completion of fuzzy metric spaces. Fuzzy Sets and Systems 130 (2002), 399-404.   CrossRef
  13. V. Gregori and S. Romaguera: Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems 144 (2004), 411-420.   CrossRef
  14. V. Gregori, S. Morillas and A. Sapena: Examples of fuzzy metrics and applications. Fuzzy Sets and Systems 170 (2011), 95-111.   CrossRef
  15. F. Hausdorff: Set Theory. Chelsea, New York 1957.   CrossRef
  16. N.V. Hop: Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures. Inform. Sci. 177 (2007), 1977-1991.   CrossRef
  17. N. V. Hop: Solving linear programming problems under fuzziness and randomness environment using attainment values. Inform. Sci. 177 (2007), 2971-2984.   CrossRef
  18. S. Y. Joo and Y. K. Kim: The Skorokhod topology on space of fuzzy numbers. Fuzzy Sets and Systems 111 (2000), 497-501.   CrossRef
  19. S. Y. Joo and Y. K. Kim: Topological properties on the space of fuzzy sets. J. Math. Anal. Appl. 246 (2000), 576-590.   CrossRef
  20. O. Kaleva and S. Seikkala: On fuzzy metric spaces. Fuzzy Sets and Systems 12 (1984), 215-229.   CrossRef
  21. O. Kaleva: On the convergence of fuzzy sets. Fuzzy Sets and Systems 17 (1985), 53-65.   CrossRef
  22. O. Kaleva: Fuzzy differential equations. Fuzzy Sets and Systems 24 (1987), 301-317.   CrossRef
  23. Y. K. Kim: Compactness and convexity on the space of fuzzy sets. J. Math. Anal. Appl. 264 (2001), 122-132.   CrossRef
  24. Y. K. Kim: Compactness and convexity on the space of fuzzy sets II. Nonlinear Anal. Theory Methods Appl. 57 (2004), 639-653.   CrossRef
  25. E. P. Klement, R. Mesiar and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000.   CrossRef
  26. I. Kramosil and J. Michálek: Fuzzy metric and statistical metric spaces. Kybernetika 11 (1975), 326-334.   CrossRef
  27. G. Matheron: Random Sets and Integral Geometry. Wiley, New York 1975.   CrossRef
  28. B. S. Mordukhovich and Y. Shao: Fuzzy calculus for coderivatives of multifunctions. Nonlinear Anal. Theory Methods Appl. 29 (1997), 605-626.   CrossRef
  29. S. B. Nadler Jr: Multi-valued contraction mappings. Pacific J. Math. 30 (1969), 475-487.   CrossRef
  30. M. L. Puri and D. A. Ralescu: Fuzzy random variables. J. Math. Anal. Appl. 114 (1986), 409-422.   CrossRef
  31. D. Qiu and W. Zhang: On Decomposable Measures Induced by Metrics. J. Appl. Math. Volume 2012, Article ID 701206, 8 pages.   CrossRef
  32. D. Qiu and W. Zhang: The strongest t-norm for fuzzy metric spaces. Kybernetika 49 (2013), 141-148.   CrossRef
  33. D. Qiu, W. Zhang and C. Li: On decomposable measures constructed by using stationary fuzzy pseudo-ultrametrics. Int. J. Gen. Syst. 42 (2013), 395-404.   CrossRef
  34. D. Qiu, W. Zhang and C. Li: Extension of a class of decomposable measures using fuzzy pseudometrics. Fuzzy Sets and Systems 222 (2013), 33-44.   CrossRef
  35. J. Rodríguez-López and S. Romaguera: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 147 (2004), 273-283.   CrossRef
  36. S. Romaguera and M. Sanchis: On fuzzy metric groups. Fuzzy Sets and Systems 124 (2001), 109-115.   CrossRef
  37. F. G. Shi and C. Y. Zheng: Metrization theorems in L-topological spaces. Fuzzy Sets and Systems 149 (2005), 455-471.   CrossRef
  38. E. Trillas: On the use of words and fuzzy sets. Inf. Sci. 176 (2006), 1463-1487.   CrossRef
  39. L. A. Zadeh: Fuzzy sets. Inform. Control 8 (1965), 338-353.   CrossRef
  40. W. Zhang, D. Qiu, Z. Li and G. Xiong: Common fixed point theorems in a new fuzzy metric space. J. Appl. Math. Volume 2012, Article ID 890678, 18 pages.   CrossRef