Fuzzy b-Metric Spaces

  • Sorin Nădăban Department of Mathematics and Computer Science Aurel Vlaicu University of Arad, Elena Drăgoi 2, RO-310330 Arad, Romania

Abstract

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step.

References

[1] Alghamdi, M.A., Hussain, N., Salimi, P. (2013); Fixed point and coupled fixed point theorems on b-metric-like spaces, Journal of Inequalities and Applications, 2013:402.

[2] Amini-Harandi, A. (2012). Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory and Applications, 2012:204.
http://dx.doi.org/10.1186/1687-1812-2012-204

[3] Bag, T. (2013); Fuzzy cone metric spaces and fixed point theorems of contractive mappings, Annals of Fuzzy Mathematics and Informatics, 6(3): 657–668.

[4] Bag, T. (2014); Some fixed point theorems in fuzzy cone b-metric spaces, International Journal of Fuzzy Mathematics and Systems, 4(2): 255–267.

[5] Bakhtin, I.A. (1989); The contraction mapping principle in quasi-metric spaces, Funct. Anal. Unianowsk Gos. Ped. Inst., 30: 26–37.

[6] Boriceanu, M., Bota, M., Petruşel, A. (2010); Multivalued fractals in b-metric spaces, Central European Journal of Mathematics, 8(2): 367–377.
http://dx.doi.org/10.2478/s11533-010-0009-4

[7] Boriceanu, M., Petruşel, A., Rus, I.A. (2010); Fixed point theorems for some multivalued generalized contraction in b-metric spaces, International J. Math. Statistics, 6: 65–76.

[8] Boriceanu, M. (2009); Strict fixed point theorems for multivalued operators in b-metric spaces, Intern. J. Modern Math., 4: 285–301.

[9] Chifu, C., Petruşel, G. (2014); Fixed point for multivalued contraction in b-metric spaces with applications to fractals, Taiwanese Journal of Mathematics, 18(5): 1365–1375.

[10] Czerwik, S. (1993); Contraction mappings in b-metric space, Acta Math. Inf. Univ. Ostraviensis, 1: 5–11.

[11] Czerwik, S. (1998); Non-linear set-valued contraction mappings in b-metric spaces, Atti. Sem. Math. Fig. Univ. Modena, 46(2): 263–276.

[12] George, A., Veeramani, P. (1994); On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64: 395–399.
http://dx.doi.org/10.1016/0165-0114(94)90162-7

[13] Gregori, V., Romaguera, S. (2004); Fuzzy quasi-metric spaces, Applied General Topology, 5(1): 128–136.

[14] Hussain, N., Shah, M.H. (2011); KKM mappings in cone b-metric spaces, Comput. Math. Appl., 61(4): 1677–1684.

[15] Kaleva, O., Seikkala, S. (1984); On fuzzy metric spaces, Fuzzy Sets and Systems, 12: 215– 229.
http://dx.doi.org/10.1016/0165-0114(84)90069-1

[16] Kramosil, I., Michálek, J. (1975); Fuzzy metric and statistical metric spaces, Kybernetica, 11: 326–334.

[17] Matthews, S.G. (1994); Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, Ann. New York Acad. Sci., Vol. 728, The New York Academy of Sciences, 183–197.
http://dx.doi.org/10.1111/j.1749-6632.1994.tb44144.x

[18] Nădăban, S. (2015); Fuzzy euclidean normed spaces for data mining applications, International Journal of Computers Communications & Control, 10(1): 70–77.
http://dx.doi.org/10.15837/ijccc.2015.1.1564

[19] Schweizer, B., Sklar, A. (1960); Statistical metric spaces, Pacific J. Math., 10: 314–334.

[20] Shah, M.H., Hussain, N. (2012); Nonlinear contraction in partially ordered quasi b-metric spaces, Commun. Korean Math. Soc., 27(1): 117–128.
http://dx.doi.org/10.4134/CKMS.2012.27.1.117

[21] Shatanawi, W., Pitea, A., Lazović, R. (2014); Contraction conditions using comparison function on b-metric spaces, Fixed Point Theory and Applications, 2014:135.

[22] Singh, S.L., Prasad, B. (2008); Some coincidence theorems and stability of iterative procedures, Computers and Mathematics with Applications, 55: 2512–2520.
http://dx.doi.org/10.1016/j.camwa.2007.10.026

[23] Zadeh, L.A. (1965); Fuzzy Sets, Informations and Control, 8: 338–353.
http://dx.doi.org/10.1016/S0019-9958(65)90241-X
Published
2016-01-26
How to Cite
NĂDĂBAN, Sorin. Fuzzy b-Metric Spaces. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL, [S.l.], v. 11, n. 2, p. 273-281, jan. 2016. ISSN 1841-9844. Available at: <http://univagora.ro/jour/index.php/ijccc/article/view/2443>. Date accessed: 02 july 2020. doi: https://doi.org/10.15837/ijccc.2016.2.2443.

Keywords

Fuzzy b-metric spaces, fuzzy quasi-b-metric, fuzzy quasi-pseudo-bmetric, b-metric space