The relationship between modular metrics and fuzzy metrics revisited

Yıl 2024, , 90 - 97, 15.09.2024

Salvador Romaguera Bonilla

https://doi.org/10.33205/cma.1502096

Öz

In a famous article published in 1975, Kramosil and Mich\'{a}lek introduced a notion of fuzzy metric that was the origin of numerous researches and publications in several frameworks and fields. In 2010, Chistyakov introduced and discussed in detail the concept of modular metric. Since then, some authors have investigated the problem of establishing connections between the notions of fuzzy metric and modular metric, obtaining positive partial solutions. In this paper, we are interested in determining the precise relationship between these two concepts. To achieve this goal, we examine a proof, based on the use of uniformities, of the important result that the topology induced by a fuzzy metric is metrizable. As a consequence of that analysis, we introduce the notion of a weak fuzzy metric and show that every weak fuzzy metric, with continuous t-norm the minimum t-norm, generates a modular metric and, conversely, we show that every modular metric generates a weak fuzzy metric, with continuous t-norm the product t-norm. It follows that every modular metric can be generated from a suitable weak fuzzy metric, and that several examples and properties of modular metrics can be directly deduced from those previously obtained in the field of fuzzy metrics.

Anahtar Kelimeler

fuzzy metric, weak fuzzy metric, modular metric

Kaynakça

• C. Alegre, S. Romaguera: Characterizations of metrizable topological vector spaces and their asymmetric generalizations in terms of fuzzy (quasi-)norms, Fuzzy Sets Syst., 161 (2010), 2181–2192.
• C. di Bari, C. Vetro: A fixed point theorem for a family of mappings in a fuzzy metric space, Rend. Circolo Mat. Palermo, 52 (2003), 315–321.
• Y.J. Cho, M. Grabiec and V. Radu: On Nonsymmetric Topological and Probabilistic Structures, Nova Science Publisher, Inc., New York (2006).
• V. V. Chistyakov: Modular metric spaces, I: basic concepts, Nonlinear Anal. TMA, 72 (2010), 1–14.
• V. V. Chistyakov: Metric Modular Spaces Theory and Applications, SpringerBriefs in Mathematics, Cham (2015).
• A. George, P. Veeramani: On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395–399.
• D. Gopal, J. Martínez-Moreno: Recent Advances and Applications of Fuzzy Metric Fixed Point Theory, CRC Press, Taylor & Francis Group, Boca Raton (2024).
• M. Grabiec: Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (1989), 385–389.
• V. Gregori, S. Romaguera: Some properties of fuzz metric spaces, Fuzzy Sets Syst., 115 (2000), 485–489.
• O. Hadzic, E. Pap: Fixed Point Theory in Probabilistic Metric Spaces Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (2001).
• N. Hussain, P. Salimi: Implicit contractive mappings in modular metric and fuzzy metric spaces, Sci. World J., 2014 (2014), Article ID:981578.
• E. Klement, R. Mesiar and E. Pap: Triangular Norms, Kluwer Academic, Dordrecht (2000).
• I. Kramosil, J. Michálek: Fuzzy metrics and statistical metric spaces, Kybernetika, 11 (1975), 326–334.
• J. Musielak, W. Orlicz: On modular spaces, Studia Math., 18 (1959), 49–65.
• H. Nakano: Modulared Semi-Ordered Linear Spaces, in: Tokyo Math. Book Ser., vol. 1, Maruzen Co., Tokyo (1950).
• H. Nakano; Topology and Linear Topological Spaces, in: Tokyo Math. Book Ser., vol. 3, Maruzen Co., Tokyo (1951).
• W. Orlicz: A note on modular spaces, I, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys., 9 (1961), 157–162.
• V. Radu: On the relationship between locally K-convex spaces and random normed spaces over valued fields, Seminarul de Teoria Probabilitatilor, STPA, West University of Timisoara, Vol. 37 (1978).
• B. Schweizer, A. Sklar and E. Thorp: The metrization of statistical metric spaces, Pacific J. Math., 10 (1960), 673–675.
• F. Tchier, C. Vetro and F. Vetro: From fuzzy metric spaces to modular metric spaces: a fixed point approach, J. Nonlinear Sci. Appl., 10 (2017), 451–464.