Honam Mathematical Journal (호남수학학술지)

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STATISTICALLY LOCALIZED SEQUENCES IN 2-NORMED SPACES

  • Received : 2019.07.11
  • Accepted : 2019.10.04
  • Published : 2020.03.25
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Abstract

We introduce statistically localized sequences in 2-normed spaces and give some main properties of statistically localized sequences. Also, we prove that a sequence is statistically Cauchy sequence if and only if its statistical barrier is equal to zero. Moreover, we define the uniformly statistically localized sequences on 2-normed spaces and investigate its relationship with statistically Cauchy sequences.

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References

  1. C. Belen and S. A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219(18) (2013), 9821-9826. https://doi.org/10.1016/j.amc.2013.03.115
  2. M. Basarir, S. Konca, E. E. Kara, Some generalized difference statistically convergent sequence spaces in 2 -normed space, J. Inequal. Appl. 2013(177) (2013), pp. 10. https://doi.org/10.1186/1029-242X-2013-10
  3. B. J. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis 8 (1988), 47-63. https://doi.org/10.1524/anly.1988.8.12.47
  4. O. H. H. Edely, S. A. Mohiuddine and A. K. Noman, Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett. 23 (2010), 1382-1387. https://doi.org/10.1016/j.aml.2010.07.004
  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, Pacific J. Math. 95(2) (1981), 293-305. https://doi.org/10.2140/pjm.1981.95.293
  7. J. A. Fridy, On statistical convergence, Analysis (Munich) 5 (1985), 301-313.
  8. S. Gahler, 2-metrische Raume und ihre topologische Struktur, Math. Nachr. 26 (1963), 115-148. https://doi.org/10.1002/mana.19630260109
  9. M. Gurdal and S. Pehlivan, The statistical convergence in 2-normed spaces, Southeast Asian Bull. Math. 33(2) (2009), 257-264.
  10. M. Gurdal and I. Acik, On I-cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11(2) (2008), 349-354.
  11. M. Gurdal and U. Yamanci, Statistical convergence and some questions of operator theory, Dynam. Syst. Appl. 24 (2015), 305-312.
  12. U. Kadak and S. A. Mohiuddine, Generalized Statistically Almost Convergence Based on the Difference Operator Which Includes the (p, q)-Gamma Function and Related Approximation Theorems, Results Math. 73(1) (2018), Article 9.
  13. O. Kisi and E. Dundar, Rough ${\tau}_2$-lacunary statistical convergence of double sequences, J. Inequal. Appl. 2018 (2018), Article 230.
  14. L. N. Krivonosov, Localized sequences in metric spaces, Soviet Math. (Iz. VUZ), 18(4) (1974), 37-44.
  15. S. A. Mohiuddine, H. Sevli and M. Cancan, Statistical convergence in fuzzy 2-normed space, J. Comput. Anal. Appl. 12(4) (2010), 787-798.
  16. S. A. Mohiuddine and M. Aiyub, Lacunary statistical convergence in random 2-normed spaces, Appl. Math. Inf. Sci. 6(3) (2012), 581-585.
  17. S. A. Mohiuddine, Statistical weighted A-summability with application to Korovkin's type approximation theorem, J. Inequal. Appl. 2016 (2016), Article 101.
  18. 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 Fs. Nat. Ser. A Math. RACSAM 113 (2019), 1955-1973.
  19. S. A. Mohiuddinea, 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(5) (2019), 492-506. https://doi.org/10.1080/03081079.2019.1608985
  20. M. Mursaleen, On statistical convergence in random 2-normed spaces, Acta Sci. Math. (Szeged) 76 (2010), 101-109.
  21. M. Mursaleen and S. A. Mohiuddine, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos, Solitons and Fractals 41 (2009), 2414-2421. https://doi.org/10.1016/j.chaos.2008.09.018
  22. A. A. Nabiev, E. Savas and M. Gurdal, Statistically localized sequences in metric spaces, J. App. Anal. Comp. 9(2) (2019), 739-746.
  23. A. A. Nabiev, E. Savas and M. Gurdal, I-localized sequences in metric spaces, Facta Univ. Ser. Math. Inform. (2019), (to appear).
  24. F. Nuray, U. Ulusu and E. Dundar, Lacunary statistical convergence of double sequences of sets, Soft Computing 20(7) (2016), 2883-2888. https://doi.org/10.1007/s00500-015-1691-8
  25. D. Rath and B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math. 25(4)(1994), 381-386.
  26. W. F. Raymond and Ye. J. Cho, Geometry of linear 2-normed spaces, Huntington, N.Y. Nova Science Publishers, 2001.
  27. E. Savas and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), 826-830. https://doi.org/10.1016/j.aml.2010.12.022
  28. R. Savas and S. A. Sezer, Tauberian theorems for sequences in 2-normed spaces, Results Math. 72 (2017), 1919-1931. https://doi.org/10.1007/s00025-017-0747-8
  29. 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
  30. U. Yamanci and M. Gurdal, Statistical convergence and operators on Fock space, New York J. Math. 22 (2016), 199-207.
  31. S. Yegul and E. Dundar, On statistical convergence of sequences of functions in 2-normed spaces, J. Classical Anal. 10(1) (2017), 49-57. https://doi.org/10.7153/jca-10-05
  32. S. Yegul and E. Dundar, Statistical convergence of double sequences of functions and some properties in 2-normed spaces, Facta Univ. Ser. Math. Inform. 33(5) (2018), 705-719.