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Some observations on ℐ-statistically pre-Cauchy sequences of complex uncertain variables defined by Orlicz functions

  • Binod Chandra Tripathy ORCID logo , Ömer Kişi ORCID logo EMAIL logo and Birojit Das ORCID logo
Published/Copyright: May 15, 2024

Abstract

In this research article, we introduce -statistically pre-Cauchy sequences of complex uncertain variables in five different aspects of uncertainty, namely: in mean, in measure, in distribution, in almost sure, and in uniformly almost sure. We also explore the connection between -statistically pre-Cauchy sequences and -statistically convergent sequences using complex uncertain variables. Additionally, we initiate the study of -statistically pre-Cauchy sequences of complex uncertain variables through Orlicz functions.

MSC 2020: 40A35; 03E72

Acknowledgements

The authors would like to thank the Editor-in-Chief and the anonymous reviewers for their insightful suggestions and careful reading of the manuscript.

References

[1] P. Baliarsingh, On statistical deferred A-convergence of uncertain sequences, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 29 (2021), no. 4, 499–515. 10.1142/S0218488521500215Search in Google Scholar

[2] C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013), no. 18, 9821–9826. 10.1016/j.amc.2013.03.115Search in Google Scholar

[3] X. Chen, Y. Ning and X. Wang, Convergence of complex uncertain sequences, J. Intell. Fuzzy Syst. 30 (2016), no. 6, 3357–3366. 10.3233/IFS-152083Search in Google Scholar

[4] J. Connor, J. Fridy and J. Kline, Statistically pre-Cauchy sequences, Analysis 14 (1994), no. 4, 311–317. 10.1524/anly.1994.14.4.311Search in Google Scholar

[5] B. Das, B. C. Tripathy, P. Debnath and B. Bhattacharya, Characterization of statistical convergence of complex uncertain double sequence, Anal. Math. Phys. 10 (2020), no. 4, Paper No. 71. 10.1007/s13324-020-00419-7Search in Google Scholar

[6] B. Das, B. C. Tripathy, P. Debnath and B. Bhattacharya, Study of matrix transformation of uniformly almost surely convergent complex uncertain sequences, Filomat 34 (2020), no. 14, 4907–4922. 10.2298/FIL2014907DSearch in Google Scholar

[7] B. Das, B. C. Tripathy, P. Debnath and B. Bhattacharya, Statistical convergence of complex uncertain triple sequence, Comm. Statist. Theory Methods 51 (2022), no. 20, 7088–7100. 10.1080/03610926.2020.1871016Search in Google Scholar

[8] B. Das, B. C. Tripathy, P. Debnath, J. Nath and B. Bhattacharya, Almost convergence of complex uncertain triple sequences, Proc. Nat. Acad. Sci. India Sect. A 91 (2021), no. 2, 245–256. 10.1007/s40010-020-00721-wSearch in Google Scholar

[9] P. Das and E. Savas, On I-statistically pre-Cauchy sequences, Taiwanese J. Math. 18 (2014), no. 1, 115–126. 10.11650/tjm.18.2014.3157Search in Google Scholar

[10] P. Das, E. Savas and S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett. 24 (2011), no. 9, 1509–1514. 10.1016/j.aml.2011.03.036Search in Google Scholar

[11] D. Datta and B. C. Tripathy, Double sequences of complex uncertain variables defined by Orlicz function, New Math. Nat. Comput. 16 (2020), no. 3, 541–550. 10.1142/S1793005720500325Search in Google Scholar

[12] K. Demirci, -limit superior and limit inferior, Math. Commun. 6 (2001), no. 2, 165–172. Search in Google Scholar

[13] A. J. Dutta, A. Esi and B. C. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, J. Math. Anal. 4 (2013), no. 2, 16–22. Search in Google Scholar

[14] A. J. Dutta and B. C. Tripathy, Statistically pre-Cauchy fuzzy real-valued sequences defined by Orlicz function, Proyecciones 33 (2014), no. 3, 235–243. 10.4067/S0716-09172014000300001Search in Google Scholar

[15] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Search in Google Scholar

[16] B. Hazarika, A. Alotaibi and S. A. Mohiuddine, Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Computing 24 (2020), 6613–6622. 10.1007/s00500-020-04805-ySearch in Google Scholar

[17] V. A. Khan and S. Ashadul Rahaman, Intuitionistic fuzzy Tribonacci I-convergent sequence spaces, Math. Slovaca 72 (2022), no. 3, 693–708. 10.1515/ms-2022-0047Search in Google Scholar

[18] V. A. Khan, K. Ebadullah and A. Ahmad, I-pre-Cauchy sequences and Orlicz functions, J. Math. Anal. 3 (2012), no. 1, 21–26. Search in Google Scholar

[19] V. A. Khan and Q. M. D. Lohani, Statistically pre-Cauchy sequences and Orlicz functions, Southeast Asian Bull. Math. 31 (2007), no. 6, 1107–1112. Search in Google Scholar

[20] V. A. Khan, N. Khan, A. Esi and S. Tabassum, -pre-Cauchy double sequences and Orlicz functions, Eng. Sci. Res. 5 (2013), no. 5A, 52–56. 10.4236/eng.2013.55A008Search in Google Scholar

[21] V. A. Khan and S. Tabassum, Statistically pre-Cauchy double sequences, Southeast Asian Bull. Math. 36 (2012), no. 2, 249–254. Search in Google Scholar

[22] P. Kostyrko, T. Šalát and W. Wilczyński, -convergence, Real Anal. Exchange 26 (2000/01), no. 2, 669–685. 10.2307/44154069Search in Google Scholar

[23] B. K. Lahiri and P. Das, Further results on I-limit superior and limit inferior, Math. Commun. 8 (2003), no. 2, 151–156. Search in Google Scholar

[24] B. Liu, Uncertainty Theory, 2nd ed., Stud. Fuzziness Soft Comput. 154, Springer, Berlin, 2007. Search in Google Scholar

[25] S. A. Mohiuddine, A. Asiri and B. Hazarika, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst. 48 (2019), no. 5, 492–506. 10.1080/03081079.2019.1608985Search in Google Scholar

[26] S. A. Mohiuddine, B. Hazarika and A. Alotaibi, On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Syst. 32 (2017), 4331–4342. 10.3233/JIFS-16974Search in Google Scholar

[27] S. A. Mohiuddine, H. Şevli and M. Cancan, Statistical convergence in fuzzy 2-normed space, J. Comput. Anal. Appl. 12 (2010), no. 4, 787–798. Search in Google Scholar

[28] M. Mursaleen and S. A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62 (2012), no. 1, 49–62. 10.2478/s12175-011-0071-9Search in Google Scholar

[29] J. Nath, B. Das, B. Bhattacharya and B. C. Tripathy, On statistically pre-Cauchy sequences of complex uncertain variables defined by Orlicz functions, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 31 (2023), no. 2, 191–207. 10.1142/S0218488523500113Search in Google Scholar

[30] J. Nath, B. C. Tripathy, P. Debnath and B. Bhattacharya, Strongly almost convergence in sequences of complex uncertain variables, Comm. Statist. Theory Methods 52 (2023), no. 3, 714–729. 10.1080/03610926.2021.1921802Search in Google Scholar

[31] P. K. Nath and B. C. Tripathy, Convergent complex uncertain sequences defined by Orlicz function, An. Univ. Craiova Ser. Mat. Inform. 46 (2019), no. 1, 139–149. Search in Google Scholar

[32] Z. Peng, Complex uncertain variable, Ph.D. thesis, Tsinghua University, 2012. Search in Google Scholar

[33] D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25 (1994), no. 4, 381–386. Search in Google Scholar

[34] E. Savaş, On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function, J. Inequal. Appl. 2010 (2010), Article ID 482392. 10.1155/2010/482392Search in Google Scholar

[35] E. Savaş, A-sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function, Abstr. Appl. Anal. 2011 (2011), Article ID 741382. 10.1155/2011/741382Search in Google Scholar

[36] E. Savas and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), no. 6, 826–830. 10.1016/j.aml.2010.12.022Search in Google Scholar

[37] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. 10.1080/00029890.1959.11989303Search in Google Scholar

[38] B. C. Tripathy and S. Borgohain, Statistically convergent difference sequence spaces of fuzzy real numbers defined by Orlicz function, Thai J. Math. 11 (2013), no. 2, 357–370. Search in Google Scholar

[39] B. C. Tripathy and P. K. Nath, Statistical convergence of complex uncertain sequences, New Math. Nat. Comput. 13 (2017), no. 3, 359–374. 10.1142/S1793005717500090Search in Google Scholar

[40] B. C. Tripathy and M. Sen, On generalized statistically convergent sequences, Indian J. Pure Appl. Math. 32 (2001), no. 11, 1689–1694. Search in Google Scholar

[41] C. You, On the convergence of uncertain sequences, Math. Comput. Modelling 49 (2009), no. 3–4, 482–487. 10.1016/j.mcm.2008.07.007Search in Google Scholar

Received: 2023-12-01
Revised: 2024-02-13
Accepted: 2024-04-20
Published Online: 2024-05-15
Published in Print: 2025-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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