DOI QR코드

DOI QR Code

ON ASYMPTOTICALLY LACUNARY STATISTICAL EQUIVALENT TRIPLE SEQUENCES VIA IDEALS AND ORLICZ FUNCTION

  • Received : 2021.02.23
  • Accepted : 2021.03.19
  • Published : 2021.06.25

Abstract

In the present paper, we introduce the concepts of $\mathcal{I}$-asymptotically statistical $\tilde{\phi}$-equivalence and $\mathcal{I}$-asymptotically lacunary statistical $\tilde{\phi}$-equivalence for triple sequences. Moreover, we give the relations between these new notions.

Keywords

Acknowledgement

The authors thank to the referees for valuable comments and fruitful suggestions which enhanced the readability of the paper.

References

  1. T. Acar and S. A. Mohiuddine, Statistical (C, 1)(E, 1) summability and Korovkin's theorem, Filomat, 30(2) (2016), 387-393. https://doi.org/10.2298/FIL1602387A
  2. B. Altay and F. Basar, Some new spaces of double sequences, J. Math. Anal. Appl., 309(1) (2005), 70-90. https://doi.org/10.1016/j.jmaa.2004.12.020
  3. F. Basar, Summability Theory and its Applications, Bentham Science Publishers, Istanbul, 2012.
  4. F. Basar and H. Dutta, Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties, CRC Press, Taylor & Francis Group, Monographs and Research Notes in Mathematics, Boca Raton.London.New York, 2020.
  5. P. Das, E. Savas and S. K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Lett., 24 (2011), 1509-1514. https://doi.org/10.1016/j.aml.2011.03.036
  6. H. Dutta and F. Basar, A generalization of Orlicz sequence spaces by Cesaro mean of order one, Acta Math. Univ. Comen., 80(2) (2011), 185-200.
  7. A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. Inf. Sci., 9(5) (2015), 2529-2534.
  8. H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
  9. J. A. Fridy, On statistical convergence, Analysis (Munich), 5 (1985), 301-313.
  10. J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43-51. https://doi.org/10.2140/pjm.1993.160.43
  11. M. Gurdal and M. B. Huban, On I-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik, 66(1) (2014), 73-83.
  12. M. Gurdal, M. B. Huban and U. Yamanci, On generalized statistical limit points in random 2-normed spaces, AIP Conf. Proc., 1479(2012), 950-954.
  13. M. Gurdal, N. Sari , E. Savas, A-statistically localized sequences in n-normed spaces, Commun. Fac. Sci. Univ. Ank. S'er. A1 Math. Stat., 69(2) (2020), 1484-1497.
  14. M. Gurdal and A. Sahiner, Extremal I-limit points of double sequences, Appl. Math. E-Notes, 8(2008), 131-137.
  15. B. Hazarika, A. Alotaibi and S. A. Mohiudine, Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Comput, 24(9) (2020), 6613-6622. https://doi.org/10.1007/s00500-020-04805-y
  16. M. B. Huban and M. Gurdal, Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function, J. Class. Anal., (2021) (accepted).
  17. M.B. Huban, M. Gurdal and E. Savas, I-statistical limit superior and I-statistical limit inferior of triple sequences, 7th International Conference on Recent Advances in Pure and Applied Mathematics, Proceeding Book of ICRAPAM, (2020), 42-49.
  18. P. Kostyrko, T. Salat and W. Wilczynski, I-convergence, Real Anal. Exchange, 26 (2000-2001), 669-685. https://doi.org/10.2307/44154069
  19. J. Li, Asymptotic equivalence of sequences and summability, Internat J. Math. Math. Sci., 20(4) (1997), 749-758. https://doi.org/10.1155/S0161171297001038
  20. M. Marouf, Asymptotic equivalence and summability, Internat J. Math. Math. Sci., 16(4) (1993), 755-762. https://doi.org/10.1155/S0161171293000948
  21. S. A. Mohiuddine and B. A. S. Alamri, Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 113(3) (2019), 1955-1973. https://doi.org/10.1007/s13398-018-0591-z
  22. M. Mursaleen and F. Basar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton.London.New York, 2020.
  23. M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003) 223-231. https://doi.org/10.1016/j.jmaa.2003.08.004
  24. A. Nabiev, S. Pehlivan and M. Gurdal, On I-Cauchy sequences, Taiwanese J. Math., 11(2) (2007), 569-576. https://doi.org/10.11650/twjm/1500404709
  25. R. F. Patterson, On asymptotically statistically equivalent sequences, Demonstr. Math., 36(1) (2003), 149-153. https://doi.org/10.1515/dema-2003-0116
  26. R. F. Patterson and E. Savas, On asymptotically lacunary statistical equivalent sequences, Thai J. Math., 4(2)(2006), 267-272.
  27. M. M. Rao and Z. D. Ren, Applications of Orlicz spaces, Marcel Dekker Inc., 2002.
  28. A. Sahiner, M. Gurdal and F.K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math., 8(2) (2007), 49-55.
  29. A. Sahiner and B. C. Tripathy, Some I-related properties of triple sequences, Selcuk J. Appl. Math., 9(2) (2008), 9-18.
  30. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
  31. E. Savas and S. Debnath, Lacunary statistically ϕ-convergence, Note Mat., 39(2) (2019), 111-119.
  32. E. Savas, U. Yamanci and M. Gurdal, I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 2324-2332.
  33. R. Savas, Multiple λµ-statistically convergence via $\tilde{\phi}$-functions, Math. Methods Appl. Sci., doi:10.1002/mma.7027 (2020), 1-8.