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Licensed Unlicensed Requires Authentication Published by De Gruyter October 16, 2022

Approximity of asymmetric metric spaces

  • Sanjoy Ghosal EMAIL logo , Sourav Mandal and Mandobi Banerjee
From the journal Mathematica Slovaca

Abstract

In this present work, we perceive the ideas of rough weighted statistical limit set as well as rough weighted statistical cluster points set and originate these conceptions into asymmetric metric spaces. On this context we frame out several results which substantially intensify these perceptions. While explicating such notions in terms of their asymmetric concepts, this generalization despite unfollows some previous results rather generates new characteristics. Also, we will adorn a sufficient condition using rough weighted statistical convergence which converts asymmetric metric spaces to approximate metric spaces.


Research of the second author is supported by UGC Research, HRDG, India.



banerjeeju@rediffmail.com
  1. (Communicated by L’ubica Holá)

Acknowledgement

Thankful to the Editor and Referees for their several valuable suggestions.

References

[1] Aizpuru, A.—Listán-García, M. C.—Rambla-Barreno, F.: Density by moduli and statistical convergence, Quaest. Math. 37(4) (2014), 525–530.Search in Google Scholar

[2] Arslan, M.—Dündar, E.: Rough convergence in 2-normed spaces, Bull. Math. Anal. Appl. 10(3) (2018), 1–9.Search in Google Scholar

[3] Arslan, M.—Dündar, E.: On rough convergence in 2-normed spaces and some properties, Filomat 33(16) (2019), 5077–5086.Search in Google Scholar

[4] Aytar, S.: The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optim. 29(3–4) (2008), 283–290.Search in Google Scholar

[5] Aytar, S.: Rough statistical convergence, Numer. Funct. Anal. Optim. 29(3–4) (2008), 291–303.Search in Google Scholar

[6] Aytar, S.: Rough statistical cluster points, Filomat 31(16) (2017), 5295–5304.Search in Google Scholar

[7] Cinar, M.—Et, M.: Generalized weighted statistical convergence of double sequences and applications, Filomat 30(3) (2016), 753—762.Search in Google Scholar

[8] Collins, J.—Zimmer, J.: An asymmetric Arzelà-Ascoli theorem, Topology Appl. 154 (2007), 2312–2322.Search in Google Scholar

[9] Connor, J.—Kline, J.: On statistical limit points and the consistency of ststistical convergence, J. Math. Anal. Appl. 197 (1996), 392–399.Search in Google Scholar

[10] Das, P.—Ghosal, S.—Ghosh, A.—Som, S.: Characterization of rough weighted statistical limit set, Math. Slovaca 68(4) (2018), 881–896.Search in Google Scholar

[11] Das, P.—Ghosal, S.—Pal, S.: Extending asymmetric convergence and Cauchy condition using ideals, Math. Slovaca 63(3) (2013), 545–562.Search in Google Scholar

[12] Di Maio, G.—Kočinac, L. D. R.: Statistical convergence in topology, Topology Appl. 156 (2008), 28–45.Search in Google Scholar

[13] Dündar, E.: On rough 𝓘2-convergence of double sequences, Numer. Funct. Anal. Optim. 37(4) (2016), 480–491.Search in Google Scholar

[14] Dündar, E.—Çakan, C.: Rough 𝓘-convergence, Demonstr. Math. 47(3) (2014), 638-651.Search in Google Scholar

[15] Dündar, E.—Çakan, C.: Rough convergence of double sequences, Gulf J. Math. 2(1) (2014), 45–51.Search in Google Scholar

[16] Fast, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.Search in Google Scholar

[17] Fridy, J. A.: On statistical convergence, Analysis 5(4) (1985), 301–313.Search in Google Scholar

[18] Fridy, J. A.: Statistical limit point, Proc. Amer. Math. Soc. 118(4) (1993), 1187–1192.Search in Google Scholar

[19] Ghosal, S.: Generalized weighted random convergence in probability, Appl. Math. Comput. 249 (2014), 502–509.Search in Google Scholar

[20] Ghosal, S.—Banerjee, M.: Effect of rough 𝓘-lacunary statistical convergence to induce the weighted sequence, Filomat 32(10) (2018), 3557–3568.Search in Google Scholar

[21] Ghosal, S.—Banerjee, M.: Rough weighted statistical convergence on locally solid Riesz spaces, Positivity 25 (2021), 1789–1804.Search in Google Scholar

[22] Ghosal, S.—Ghosh, A.: When deviation happens between rough statistical convergence and rough weighted statistical convergence, Math. Slovaca 69(4) (2019), 871–890.Search in Google Scholar

[23] Ghosal, S.—Ghosh, A.: Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space, Math. Slovaca 70(3) (2020), 667–680.Search in Google Scholar

[24] Ghosal, S.—Mandal, S.: Rough weighted 𝓘-αβ-statistical convergence in locally solid Riesz spaces, J. Math. Anal. Appl. 506 (2022), Art. ID 125681.Search in Google Scholar

[25] Kişi, Ü.—Dündar, E.: Rough 𝓘2-lacunary statistical convergence of double sequences, J. Inequal. Appl. 2018 (2018), Art. No. 230.Search in Google Scholar

[26] Listán-Garcia, M. C.—Rambla-Barreno, F.: A characterization of uniform rotundity in every direction in terms of rough convergence, Numer. Funct. Anal. Optim. 32(11) (2011), 1166–1174.Search in Google Scholar

[27] Listán-Garcia, M. C.—Rambla-Barreno, F.: Rough convergence and Chebyshev centers in Banach spaces, Numer. Funct. Anal. Optim. 35(4) (2014), 432–442.Search in Google Scholar

[28] Mennucci, A.: On Asymmetric Distances, Technical report, Scuola Normale Superiore, Pisa, 2004.Search in Google Scholar

[29] Mursaleen, M.—Karakaya, V.—Ertürk, M.—Gürsoy, F.: Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218(18) (2012), 9132–9137.Search in Google Scholar

[30] Nuray, F.—Ruckle, W. H.: Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513–527.Search in Google Scholar

[31] Ülmez, Ü.—Aytar, S.: The relation between rough Wijsman convergence and asymptotic cones, Turkish J. Math. 40(6) (2016), 1349–1355.Search in Google Scholar

[32] Pal, S.—Chandra, D.—Dutta, S.: Rough ideal convergence, Hacet. J. Math. Stat. 42(2) (2013), 633–640.Search in Google Scholar

[33] Phu, X. H.: Rough convergence in normed linear spaces, Numer. Funct. Anal. Optim. 22(1–2) (2001), 199–222.Search in Google Scholar

[34] Phu, X. H.: Rough convergence in infinite dimensional normed spaces, Numer. Funct. Anal. Optim. 24(3–4) (2003), 285–301.Search in Google Scholar

[35] Şalát, T.: On statistically convergent sequences of real numbers, Math. Slovaca 30(2) (1980), 139–150.Search in Google Scholar

[36] Schoenberg, I. J.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.Search in Google Scholar

[37] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74.Search in Google Scholar

[38] Tripathy, B. C.: On statistically convergent and statistically bounded sequences, Bull. Malays. Math. Soc. (Second Series) 20 (1997), 31–33.Search in Google Scholar

[39] Wilson, W. A.: On quasi-metric spaces, Amer. J. Math. 53(3) (1931), 675–684.Search in Google Scholar

Received: 2021-01-16
Accepted: 2021-09-08
Published Online: 2022-10-16
Published in Print: 2022-10-26

© 2022 Mathematical Institute Slovak Academy of Sciences

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