On ρ-statistical convergence in gradual normed linear spaces

Shyamal Debnath; Santonu Debnath; Chiranjib Choudhury

doi:10.14244/lajm.v5i1.94

Authors

Shyamal Debnath Tripura University https://orcid.org/0000-0003-2804-6564
Santonu Debnath Dhamma Dipa International Buddhist University https://orcid.org/0000-0002-4586-0691
Chiranjib Choudhury Tripura University https://orcid.org/0000-0002-5607-9884

DOI:

https://doi.org/10.14244/lajm.v5i1.94

Keywords:

Statistical convergence, ρ-statistical convergence, gradual normed linear spaces, ρ-statistical boundedness

Abstract

In this article, we present the notion of $\rho$-statistical convergence in the context of gradual normed linear spaces and explore connections between statistical boundedness in gradual normed linear spaces and $\rho$-statistical boundedness. Additionally, we introduce the concept of $\rho$-statistical Cauchy sequences in gradual normed spaces and establish the correlation between gradual $\rho$-statistical convergence and gradual $\rho$-statistical Cauchy sequences.

Author Biographies

Shyamal Debnath, Tripura University

Professor, Dept of Mathematics

Santonu Debnath, Dhamma Dipa International Buddhist University

Assistant Professor

Dept. of Mathematics

Chiranjib Choudhury, Tripura University

Ph.D Student (Degree Awarded), Dept. of Mathematics

References

[1] C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), no. 18, 9588–9594. DOI: https://doi.org/10.1016/j.amc.2013.03.115

[2] V. K. Bhardwaj, S. Dhawan and S. Gupta, Density by moduli and statistical boundedness, Abstr. Appl. Anal. 2016 (2016). DOI: https://doi.org/10.1155/2016/2143018

[3] V. K. Bhardwaj and S. Gupta, On some generalizations of statistical boundedness, J. Inequal. Appl. 2014 (2014), no. 12, 11 pp. DOI: https://doi.org/10.1186/1029-242X-2014-12

[4] H. Cakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (1995), no. 2,113–119.

[5] H. Cakalli,A study on statistical convergence, Funct. Anal. Approx. Comput. 1(2009), no. 2, 19–24.

[6] H. Cakalli,A variation on statistical ward continuity, Bull. Malays. Math. Sci. Soc. 40(2017), no. 4, 1701–1710. DOI: https://doi.org/10.1007/s40840-015-0195-0

[7] C. Choudhury and S. Debnath, On I-convergence of sequences in gradual normed linear spaces, Facta Univ. Ser. Math. Inform. 36 (2021), no. 3, 595–604. DOI: https://doi.org/10.22190/FUMI210108044C

[8] C. Choudhury and S. Debnath, On quasi statistical convergence in gradual normed linear spaces, TWMSJ.App.Eng. Math.13 (2023), no.4, 1327–1336.

[9] C. Choudhury and S. Debnath, On $lambda$-statistical convergence of sequences in gradual normed spaces, Appl. Math. E-Notes 23 (2023), 209–219.

[10] D. Dubois and H. Prade, Gradual elements in a fuzzy set, Soft Comput. 12 (2008), no. 2, 165–175. DOI: https://doi.org/10.1007/s00500-007-0187-6

[11] A. Esi, S. Araci and M. Acikgoz, Statistical convergence of Bernstein operators, Appl. Math. Inf. Sci. 10 (2016), no. 6, 2083–2086. DOI: https://doi.org/10.18576/amis/100610

[12] M. Et, V. K. Bhardwaj and S. Gupta, On deferred statistical boundedness of order $alpha$, Comm. Statist. Theory Methods 51 (2022), no. 24, 8786–8798. DOI: https://doi.org/10.1080/03610926.2021.1906434

[13] M. Et, S. A. Mohiuddine and H. Şengül, On lacunary statistical boundedness of order $alpha$, Facta Univ. Ser. Math. Inform. 31 (2016), no. 3, 707–716.

[14] M. Ettefagh, F. Y. Azari and S. Etemad, On some topological properties in gradual normed spaces, Facta Univ. Ser. Math. Inform. 35 (2020), no. 3, 549–559. DOI: https://doi.org/10.22190/FUMI2003549E

[15] M. Ettefagh, S. Etemad and F. Y. Azari, Some properties of sequences in gradual normed spaces, Asian-Eur. J. Math. 13 (2020), no. 4, Art. ID 2050085, 12 pp. D DOI: https://doi.org/10.1142/S1793557120500850

[16] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), no. 3-4,241–244. DOI: https://doi.org/10.4064/cm-2-3-4-241-244

[17] J. Fortin, D. Dubois and H. Fargier, Gradual numbers and their application to fuzzy interval analysis, IEEE Trans. Fuzzy Syst. 16 (2008), no. 2, 388–402. DOI: https://doi.org/10.1109/TFUZZ.2006.890680

[18] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–314. DOI: https://doi.org/10.1524/anly.1985.5.4.301

[19] J. A. Fridy and C. Orhan, Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (1997), no.12, 3625–3631. DOI: https://doi.org/10.1090/S0002-9939-97-04000-8

[20] M. Kucukaslan and M. Yilmazturk, On deferred statistical convergence of sequences, Kyungpook Math. J. 56(2016), no.2, 357–366. DOI: https://doi.org/10.5666/KMJ.2016.56.2.357

[21] L. Lietard and D. Rocacher, Conditions with aggregates evaluated using gradual numbers, Control Cybernet. 38 (2009), no. 2, 395–417.

[22] M. Mursaleen, $lambda$-statistical convergence, Math. Slovaca 50 (2000), no. 1, 111–115.

[23] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), no. 1, 223–231. DOI: https://doi.org/10.1016/j.jmaa.2003.08.004

[24] I. Sadeqi and F. Y. Azari, Gradual normed linear space, Iran. J. Fuzzy Syst. 8 (2011), no. 5, 131–139.

[25] E. Savas, On statistically convergent double sequences of fuzzy numbers, Inf. Sci. 162 (2004), no. 3-4, 183–192. DOI: https://doi.org/10.1016/S0020-0255(03)00306-2

[26] E. Savas and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), no. 6, 826–830. DOI: https://doi.org/10.1016/j.aml.2010.12.022

[27] B. C. Tripathy, Statistically convergent double sequences, Tamkang J. Math. 34 (2003), no. 3, 231–237. DOI: https://doi.org/10.5556/j.tkjm.34.2003.314

[28] L. A. Zadeh, Fuzzy sets, Inf. Control 8 (1965), no. 3, 338–353. DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

[29] B. Baxhaku, P. Agrawal and R. Shukla, Some fuzzy Korovkin type approximation theorems via power series summability method, Soft Comput. 26 (2022), no. 21, 11501–11507. DOI: https://doi.org/10.1007/s00500-022-07429-6