Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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TRIPLE SEQUENCES IN THE TOPOLOGY INDUCED BY RANDOM 2-NORMS

  • GURDAL, VERDA (Department of Mathematics, Suleyman Demirel University)
  • Received : 2021.12.23
  • Accepted : 2022.02.24
  • Published : 2022.03.30
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Abstract

In this article we define and study the notions of $\mathcal{I}$-convergence and $\mathcal{I}$-Cauchy of triple sequences in the topology induced by random 2-normed spaces and prove some theorems based on them.

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References

  1. B. Altay and F. Basar, Some new spaces of double sequences, J. Math. Anal. Appl. 309 (2005), 70-90. https://doi.org/10.1016/j.jmaa.2004.12.020
  2. F. BaSar, Summability Theory and its Applications, Bentham Science Publishers, Istanbul, 2012.
  3. I.A. Demirci and M. Gurdal, On lacunary generalized statistical convergent complex uncertain triple sequence, J. Intell. Fuzzy Systems 41 (2021), 1021-1029. https://doi.org/10.3233/JIFS-202964
  4. I.A. Demirci and M. Gurdal, On lacunary statistical ϕ-convergence for triple sequences of sets via ideals, J. Appl. Math. Inform. in press, 2021.
  5. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
  6. A.R. Freedman and J.J. Sember, Densities and summability, Pasific J. Math. 95 (1981), 293-305. https://doi.org/10.2140/pjm.1981.95.293
  7. J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  8. S. Gahler, 2-metrische raume und ihre topologische struktur, Math. Nachr. 26 (1963), 115-148. https://doi.org/10.1002/mana.19630260109
  9. I. Golet, On probabilistic 2-normed spaces, Novi Sad. J. Math. 35 (2005), 95-102.
  10. M. Gurdal and M.B. Huban, On I-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik 66 (2014), 73-83.
  11. M. Gurdal and S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai. J. Math. 2 (2004), 107-113.
  12. M. Gurdal and I. Acik, On I-cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11 (2008), 349-354.
  13. M.B. Huban and M. Gurdal, Wijsman lacunary invariant statistical convergence for triple sequences via Orlicz function, J. Class. Anal. 17 (2021), 119-128.
  14. M.B. Huban, M. Gurdal and H. Bayturk, On asymptotically lacunary statistical equivalent triple sequences via ideals and Orlicz function, Honam Math. J. 43 (2021), 343-357. https://doi.org/10.5831/HMJ.2021.43.2.343
  15. P. Kostyrko, M. Macaj and T. Salat, I-convergence, Real Anal. Exchange 26 (2000), 669-686. https://doi.org/10.2307/44154069
  16. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 28 (1942), 535-537. https://doi.org/10.1073/pnas.28.12.535
  17. S.A. Mohiuddine and M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inf. Sci. 6 (2012), 581-585.
  18. S.A. Mohiuddine, A. Alotaibi and S.M. Alsulami, Ideal convergence of double sequences in random 2-normed spaces, Adv. Differential Equations 2012:149 (2012), 1-8.
  19. S.A. Mohiuddine and E. Savas, Lacunary statistically convergent double sequences in probabilistic normed spaces, Ann. Univ. Ferrara Sez. VII Sci. Mat. 58 (2012), 331-339. https://doi.org/10.1007/s11565-012-0157-5
  20. M. Mursaleen, On statistically convergence in random 2-normed spaces, Acta Sci. Math. (Sezeged) 76 (2010), 101-109. https://doi.org/10.1007/BF03549823
  21. M. Mursaleen and A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca 61 (2011), 933-940. https://doi.org/10.2478/s12175-011-0059-5
  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. M. Mursaleen and S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Rep. (Bucur.) 12 (2010), 359-371.
  25. M. Mursaleen and S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62 (2012). 49-62. https://doi.org/10.2478/s12175-011-0071-9
  26. A. Nabiev, S. Pehlivan and M. Gurdal, On I-Cauchy sequences, Taiwanese J. Math. 11 (2007), 569-576. https://doi.org/10.11650/twjm/1500404709
  27. M.R.S. Rahmat and K.K. Harikrishnan, On I-convergence in the topology induced by probabilistic norms, European J. Pure Appl. Math. 2 (2009), 195-212.
  28. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
  29. E. Savas and S.A. Mohiuddine, λ-statistically convergent double sequences in probabilistic normed spaces, Math. Slovaca 62 (2012), 99-108. https://doi.org/10.2478/s12175-011-0075-5
  30. B. Schweizer and A. Sklar, Statistical metric spaces, Pacific J. Math. 10 (1960), 313-334. https://doi.org/10.2140/pjm.1960.10.313
  31. B. Schweizer and A. Sklar, Probabilistic metric spaces, North Holland, New York-Amsterdam-Oxford, 1983.
  32. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73-74. https://doi.org/10.4064/cm-2-2-98-108
  33. N. Subramanian and A. Esi, Wijsman rough lacunary statistical convergence on I Cesaro triple sequences, Int. J. Anal. Appl. 16 (2018), 643-653.
  34. A. Sahiner, M. Gurdal and F.K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math. 8 (2007), 49-55.
  35. A. Sahiner, M. Gurdal and S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007), 1477-1484. https://doi.org/10.11650/twjm/1500404879
  36. A. Sahiner and B.C. Tripathy, Some I-related properties of triple sequences, Selcuk J. Appl. Math. 9 (2008), 9-18.