On generalizations of fuzzy metric spaces

Shi, Yi ; Yao, Wei

Kybernetika, Tome 59 (2023) no. 6, pp. 880-903

Abstract (VO)

The aim of the paper is to present three-variable generalizations of fuzzy metric spaces in sense of George and Veeramani from functional and topological points of view, respectively. From the viewpoint of functional generalization, we introduce a notion of generalized fuzzy 2-metric spaces, study their topological properties, and point out that it is also a common generalization of both tripled fuzzy metric spaces proposed by Tian et al. and $\mathcal{M}$-fuzzy metric spaces proposed by Sedghi and Shobe. Since the ordinary tripled norm is the same as the ordinary norm up to the induced topology, we keep our spirit on fuzzy normed structures and introduce a concept of generalized fuzzy 2-normed spaces from the viewpoint of topological generalization. It is proved that generalized fuzzy 2-normed spaces always induces a Hausdorff topology.

Détail

DOI : 10.14736/kyb-2023-6-0880

Classification : 03B52, 03G27, 52A01
Keywords: generalized fuzzy 2-metric space; generalized fuzzy $2$-normed space; tripled fuzzy metric space; Hausdorff topology

Bibliographie

[1] Abrishami-Moghaddam, M., Sistani, T.: Some results on best coapproximation in fuzzy normed spaces. Afr. Mat. 25 (2014), 539-548. | DOI | MR

[2] Alegre, C., Romaguera, S.: Characterization of metrizable topological vector spaces and their asymmetric generalization in terms of fuzzy (quasi-)norms. Fuzzy Sets Syst. 161 (2010), 2181-2192. | DOI | MR

[3] Bag, T., Samanta, S. K.: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11 (2003), 687-705. | MR

[4] Chaipunya, P., Kumam, P.: On the distance between three arbitrary points. J. Funct. Spaces 2013 (2013), 194631. | DOI | MR

[5] Cheng, S. C., Mordeson, J. N.: Fuzzy linear operator and fuzzy normed linear spaces. Bull. Cal. Math. Soc. 86 (1994), 429-436. | MR

[6] Došenović, T., Rakić, D., Radenović, S., Carić, B.: Ćirić type nonunique fixed point theorems in the frame of fuzzy metric spaces. AIMS Math.8 (2023), 2154-2167. | DOI | MR

[7] George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64 (1994), 395-399. | DOI | MR | Zbl

[8] George, A., Veeramani, P.: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90 (1997), 365-368. | DOI | MR | Zbl

[9] Goleţ, I.: On fuzzy normed spaces. Southeast Asian Bull. Math. 31 (2007), 1-10. | DOI | MR

[10] Gregori, V., López-Crevillén, A., Morillas, S., Sapena, A.: On convergence in fuzzy metric spaces. Topology Appl. 156 (2009), 3002-3006. | DOI | MR

[11] Gregori, V., Miñana, J. J.: On fuzzy $\psi$-contractive sequences and fixed point theorems. Fuzzy Sets Syst. 300 (2016), 245-252. | DOI | MR

[12] Gregori, V., Miñana, J. J., Morillas, S., Sapena, A.: Characterizing a class of completable fuzzy metric spaces. Topology Appl. 203 (2016), 3-11. | DOI | MR

[13] Gregori, V., Miñana, J. J., Morillas, S., Miravet, D.: Fuzzy partial metric spaces. Int. J. Gen. Syst. 48 (2019), 260-279. | DOI | MR

[14] Gregori, V., Morillas, S., Sapena, A.: Examples of fuzzy metrics and applications. Fuzzy Sets Syst. 170 (2011), 95-111. | DOI | MR | Zbl

[15] Gregori, V., Morillas, S., Sapena, A.: On a class of completable fuzzy metric spaces. Fuzzy Sets Syst. 161 (2010), 2193-2205. | DOI | MR | Zbl

[16] García, J. Gutiérrez, Romaguera, S.: Examples of non-strong fuzzy metrics. Fuzzy Sets Syst. 162 (2011), 91-93. | DOI | MR

[17] Ha, K. S., Cho, Y. J., White, A.: Strictly convex and strictly $2$-convex $2$-normed spaces. Math. Jpn. 33 (1988), 3, 375-384. | DOI | MR

[18] Khan, K. A.: Generalized normed spaces and fixed point theorems. J. Math. Comput. Sci. 13 (2014), 157-167. | DOI | MR

[19] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. | MR | Zbl

[20] Kramosil, I., Michálek, J.: Fuzzy metric and statistical metric spaces. Kybernetica 11 (1975), 326-334. | DOI | MR

[21] Kundu, A., Bag, T., Nazmul, Sk.: A new generalization of normed linear space. Topol. Appl. 256 (2019), 159-176. | DOI | MR

[22] Meenakshi, A. R., Cokilavany, R.: On fuzzy $2$-normed linear spaces. J. Fuzzy Math. 9 (2001), 345-351. | MR

[23] Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. U. S. A. 28 (1942), 535-537. | DOI | MR | Zbl

[24] Merghadi, F., Aliouche, A.: A related fixed point theorem in $n$ fuzzy metric spaces. Iran. J. Fuzzy Syst. 7 (2010) 73-86. | MR

[25] Mihet, D.: Fuzzy $\psi$-contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets Syst. 159 (2008), 739-744. | DOI | MR

[26] Mohiuddine, S. A.: Some new results on approximation in fuzzy 2-normed spaces. Math. Comput. Modelling 53 (2011), 574-580. | DOI | MR

[27] Mohiuddine, S. A., Sevli, H., Cancan, M.: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl. 12 (2010), 787-798. | MR

[28] Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7 (2006), 2, 289-297. | MR

[29] Patel, U. D., Radenovic, S.: An application to nonlinear fractional differential qquation via $\alpha$-$\Gamma$F fuzzy contractive mappings in a fuzzy metric space. Mathematics 10 (16) (2022), 2831. | DOI

[30] Saadati, R., Vaezpour, S. M.: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17 (2005), 475-484. | DOI | MR

[31] Sapena, A., Morillas, S.: On strong fuzzy metrics. In: Proc. Workshop in Applied Topology WiAT09: Applied Topology: Recent progress for Computer Science, Fuzzy Math. Econom. 2009, pp. 135-141. DOI 

[32] Sedghi, S., Shobe, N.: Fixed point theorem in $\mathcal{M}$-fuzzy metric spaces with property (E). Adv. Fuzzy Math. 1 (2006), 55-65. | MR

[33] Sedghi, S., Shobe, N., Aliouche, A.: A generalization of fixed point theorems in $S$-metric spaces. Mat. Vesn. 64 (2012), 258-266. | MR

[34] Sedghi, S., Shobe, N., Zhou, H.: A common fixed point theorem in $D^*$-metric spaces. Fixed Point Theory Appl. (2007), Article ID 27906, 13 pages. | DOI | MR

[35] Sharma, A. K.: A note on fixed-points in $2$-metric spaces. Indian J. Pure Appl. Math. 11 (1980),12, 1580-1583. | MR

[36] Sharma, S., Sharma, S.: Common fixed point theorem in fuzzy $2$-metric space. Acta Cienc. Indica Math. 23 (1997), 1-4. | MR

[37] Tian, J.-F., Ha, M.-H., Tian, D.-Z.: Tripled fuzzy metric spaces and fixed point theorem. Inform. Sci. 518 (2020), 113-126. | DOI | MR

[38] Vijayabalaji, S., Thillaigovindan, N.: Fuzzy semi $n$-metric space. Bull. Pure Appl. Sci. Sect. E Math. Stat. 28 (2009), 283-293. | MR

[39] Xiao, J.-Z, Zhu, X.-H., Zhou, H.: On the topological structure of KM fuzzy metric spaces and normed spaces. IEEE Trans. Fuzzy Syst. 28 (2020), 1575-1584. | DOI

[40] Yan, C. H.: Fuzzifying topology induced by Morsi fuzzy pseudo-norms. Int. J. Gen. Syst. 51 (2022), 648-662. | DOI | MR