STATISTICAL CONVERGENCE FOR UNCERTAIN TRIPLE SEQUENCES OF FUZZY NUMBERS
Resumo
This paper introduces the notion of statistical convergence for uncertain triple sequences of
fuzzy numbers. We examine several associated types of convergence, including convergence in measure,
in mean, in almost surely, and uniformly almost surely. Moreover, illustrative examples were provided
to clarify the relationships and distinctions among these different types of convergence.
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Referências
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(2011), 2081-2093.
[2] P. Baliarsingh, S. Nanda and L. Nayak, On statistical convergence of uncertain sequence of fuzzy numbers, Int. J.
Uncertain. Fuzziness Knowledge-based Syst. 30(6) (2022), 975-989.
[3] X. Chen, Y. Ning and X. Wang, Convergence of complex uncertain sequences, J. Intell. Fuzzy Syst. 30 (2016),
3357–3366.
[4] I . C¸ anak, ¨U. Totur and Z. ¨ Onder, A Tauberian theorem for (C, 1, 1) summable double sequences of fuzzy numbers,
Iranian J. Fuzzy Syst. 14 (2017), 61-75.
[5] D. Datta and B. C. Tripathy, Convergence of complex uncertain double sequences, New Math. Nat. Comput. 16(3)
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[6] B . Das, B. C. Tripathy, P. Debnath, J. Nath and B. Bhattacharya, Almost convergence of complex uncertain triple
sequences, Proc. National Acad. Sci. India Sect. A, Phys. Sci. 91(2) (2021), 245-256.
[7] I. A. Demirci and M. G¨urdal, On lacunary generalized statistical convergent complex uncertain triple sequence, J.
Intell. Fuzzy Syst. 41(1) (2021), 1021-1029.
[8] H. Fast, Sur la convergence statistique, Colloq. Math. 2(3-4) (1951), 241-244.
[9] X. Gao, Some properties of continuous uncertain measure, Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 17
(2009), 419–426.
[10] M. B. Huban and M. G¨urdal, On deferred generalzed statistical convrergence of complex unertain triple sequences,
Palestine J. Math. 11(3) (2022), 255-266.
[11] ¨ O. Ki¸si and M. G¨urdal, Rough statistical convergence of complex uncertain triple sequence, Acta Math. Univ.
Comen. 91(4) (2022), 365-376.
[12] ¨ O. Ki¸si and C. Choudhury, Some observations on statistical convergence of uncertain double sequences of fuzzy
numbers, J. Nonlinear Sci. Appl. 17(1) (2024), 19-29.
[13] B. Liu, Uncertainty Theory, 2nd edn. Springer, Berlin (2007).
[14] B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 (2009), 3–10.
[15] B. Liu, Uncertainty theory: A branch of mathematics for modelling human uncertainty, Springer-Verlag, Berlin,
(2010).
[16] B. Liu, Why is there a need for uncertainty theory?, J. Uncertain Syst. 6 (2012), 3-10.
[17] B. Liu, Uncertainty theory, Springer-Verlag, Berlin, (2016)
[18] M. Matloka, Sequences of fuzzy numbers, Busefal, 28 (1986), 28-37.
[19] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst. 33 (1989), 123-126.
[20] J. Nath, B. C. Tripathy and B. Bhattacharya, Study on strongly almost convergence of complex uncertain triple
sequences, J. Uncertain. Syst. 14(4) (2021), 2150024.
[21] P. Kumar, V. Kumar and S. S. S Bhatia, Multiple sequences of fuzzy numbers and their statistical convergence,
Math. Sci. 6 (2012), 1-7.
[22] P. K. Nath and B. C. Tripathy, Convergent complex uncertain sequences defined by Orlicz function, Ann. Univ.
Craiova, Math. Comput. Sci. Ser. 46(1) (2019) 139–149.
[23] P. K. Nath and B. C. Tripathy, Statistical convergence of complex uncertain sequences defined by Orlicz function,
Proyecciones J. Math. 39(2) (2020), 301-315.
[24] L. Nayak, B. C. Tripathy and P. Baliarsingh, On deferred-statistical convergence of uncertain fuzzy sequences, Int.
J. Gen. Syst. 51(6) (2022), 631-647.
[25] Z. Peng, Complex uncertain variable, Doctoral Dissertation, Tsinghua University (2012).
[26] D. Rath and B. C. Tripathy, Matrix maps on sequence spaces associated with sets of integers, Indian J. Pure Appl.
Math. 27(2) (1996), 197–206.
[27] S. Roy, S. Saha and B. C. Tripathy, Some results on p-distance and sequence of complex uncertain variables,
Commun. Korean Math. Soc. 35(3) (2020), 907-916.
[28] K. Raj, S. Gorka and A. Esi, Lacunary statistical convergence of uncertain fuzzy number sequences via Orlicz
function, Math. Found. Comput. 8(6) (2025), 998-1011.
[29] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2(1) (1951), 73-74.
[30] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly,
66(5) (1959), 361-375.
[31] K. Saini and K. Raj, Applications of statistical convergence in complex uncertain sequences via deferred Riesz mean,
Int. J. Uncertain. Fuzziness Knowledge-based Syst. 29(03) (2021), 337-351.
[32] B. C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang
J. Math. 37(2) (2006), 155–162.
[33] B. C. Tripathy and P. K. Nath, Statistical convergence of complex uncertain sequences, New Math. Nat. Comput.
13(3) (2017), 359–374.
[34] B. C. Tripathy and P. J. Dowari, N¨orlund and Riesz mean of sequence of complex uncertain variables, Filomat,
32(8) (2018), 2875–2881.
[35] C. You, On the convergence of uncertain sequences, Math. Comput. Modell. 49 (2009), 482–487.
[36] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353.
(2011), 2081-2093.
[2] P. Baliarsingh, S. Nanda and L. Nayak, On statistical convergence of uncertain sequence of fuzzy numbers, Int. J.
Uncertain. Fuzziness Knowledge-based Syst. 30(6) (2022), 975-989.
[3] X. Chen, Y. Ning and X. Wang, Convergence of complex uncertain sequences, J. Intell. Fuzzy Syst. 30 (2016),
3357–3366.
[4] I . C¸ anak, ¨U. Totur and Z. ¨ Onder, A Tauberian theorem for (C, 1, 1) summable double sequences of fuzzy numbers,
Iranian J. Fuzzy Syst. 14 (2017), 61-75.
[5] D. Datta and B. C. Tripathy, Convergence of complex uncertain double sequences, New Math. Nat. Comput. 16(3)
(2020), 447-459.
[6] B . Das, B. C. Tripathy, P. Debnath, J. Nath and B. Bhattacharya, Almost convergence of complex uncertain triple
sequences, Proc. National Acad. Sci. India Sect. A, Phys. Sci. 91(2) (2021), 245-256.
[7] I. A. Demirci and M. G¨urdal, On lacunary generalized statistical convergent complex uncertain triple sequence, J.
Intell. Fuzzy Syst. 41(1) (2021), 1021-1029.
[8] H. Fast, Sur la convergence statistique, Colloq. Math. 2(3-4) (1951), 241-244.
[9] X. Gao, Some properties of continuous uncertain measure, Int. J. Uncertain. Fuzziness Knowledge-Based Syst. 17
(2009), 419–426.
[10] M. B. Huban and M. G¨urdal, On deferred generalzed statistical convrergence of complex unertain triple sequences,
Palestine J. Math. 11(3) (2022), 255-266.
[11] ¨ O. Ki¸si and M. G¨urdal, Rough statistical convergence of complex uncertain triple sequence, Acta Math. Univ.
Comen. 91(4) (2022), 365-376.
[12] ¨ O. Ki¸si and C. Choudhury, Some observations on statistical convergence of uncertain double sequences of fuzzy
numbers, J. Nonlinear Sci. Appl. 17(1) (2024), 19-29.
[13] B. Liu, Uncertainty Theory, 2nd edn. Springer, Berlin (2007).
[14] B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst. 3 (2009), 3–10.
[15] B. Liu, Uncertainty theory: A branch of mathematics for modelling human uncertainty, Springer-Verlag, Berlin,
(2010).
[16] B. Liu, Why is there a need for uncertainty theory?, J. Uncertain Syst. 6 (2012), 3-10.
[17] B. Liu, Uncertainty theory, Springer-Verlag, Berlin, (2016)
[18] M. Matloka, Sequences of fuzzy numbers, Busefal, 28 (1986), 28-37.
[19] S. Nanda, On sequences of fuzzy numbers, Fuzzy Sets Syst. 33 (1989), 123-126.
[20] J. Nath, B. C. Tripathy and B. Bhattacharya, Study on strongly almost convergence of complex uncertain triple
sequences, J. Uncertain. Syst. 14(4) (2021), 2150024.
[21] P. Kumar, V. Kumar and S. S. S Bhatia, Multiple sequences of fuzzy numbers and their statistical convergence,
Math. Sci. 6 (2012), 1-7.
[22] P. K. Nath and B. C. Tripathy, Convergent complex uncertain sequences defined by Orlicz function, Ann. Univ.
Craiova, Math. Comput. Sci. Ser. 46(1) (2019) 139–149.
[23] P. K. Nath and B. C. Tripathy, Statistical convergence of complex uncertain sequences defined by Orlicz function,
Proyecciones J. Math. 39(2) (2020), 301-315.
[24] L. Nayak, B. C. Tripathy and P. Baliarsingh, On deferred-statistical convergence of uncertain fuzzy sequences, Int.
J. Gen. Syst. 51(6) (2022), 631-647.
[25] Z. Peng, Complex uncertain variable, Doctoral Dissertation, Tsinghua University (2012).
[26] D. Rath and B. C. Tripathy, Matrix maps on sequence spaces associated with sets of integers, Indian J. Pure Appl.
Math. 27(2) (1996), 197–206.
[27] S. Roy, S. Saha and B. C. Tripathy, Some results on p-distance and sequence of complex uncertain variables,
Commun. Korean Math. Soc. 35(3) (2020), 907-916.
[28] K. Raj, S. Gorka and A. Esi, Lacunary statistical convergence of uncertain fuzzy number sequences via Orlicz
function, Math. Found. Comput. 8(6) (2025), 998-1011.
[29] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2(1) (1951), 73-74.
[30] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly,
66(5) (1959), 361-375.
[31] K. Saini and K. Raj, Applications of statistical convergence in complex uncertain sequences via deferred Riesz mean,
Int. J. Uncertain. Fuzziness Knowledge-based Syst. 29(03) (2021), 337-351.
[32] B. C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang
J. Math. 37(2) (2006), 155–162.
[33] B. C. Tripathy and P. K. Nath, Statistical convergence of complex uncertain sequences, New Math. Nat. Comput.
13(3) (2017), 359–374.
[34] B. C. Tripathy and P. J. Dowari, N¨orlund and Riesz mean of sequence of complex uncertain variables, Filomat,
32(8) (2018), 2875–2881.
[35] C. You, On the convergence of uncertain sequences, Math. Comput. Modell. 49 (2009), 482–487.
[36] L. A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353.
Publicado
2026-04-09
Seção
Special Issue: Advances in Mathematical Sciences
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