Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

ON LACUNARY STATISTICAL 𝜙-CONVERGENCE FOR TRIPLE SEQUENCES OF SETS VIA IDEALS

  • DEMIRCI, ISIL ACIK (Department of Mathematics Education, Mehmet Akif Ersoy University) ;
  • GURDAL, MEHMET (Department of Mathematics, Suleyman Demirel University)
  • Received : 2021.01.13
  • Accepted : 2022.01.24
  • Published : 2022.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In the present paper, we introduce some new notions of Wijsman ${\mathcal{I}}$-statistical convergence with the use of Orlicz function, lacunary sequence and triple sequences of sets, and obtain some analogous results from the new definitions point of views.

Keywords

References

  1. M. Baronti and P. Papini, Convergence of sequences of sets, In: Methods of functional analysis in approximation theory, ISNM, 76, Birkhauser-Verlag, Basel, 1986.
  2. R. Colak, Statistical convergence of order α, Modern methods in Analysis and its Applications, New Delhi, India, Anamaya Pub., 121-129, 2010.
  3. P. Das, E. Savas and S.K. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Letters 24 (2011), 1509-1514. https://doi.org/10.1016/j.aml.2011.03.036
  4. E. Dundar and N.P. Akin, Wijsman regularly ideal convergence of double sequences of sets, J. Intel. Fuzzy Syst. 37 (2019), 8159-8166. https://doi.org/10.3233/JIFS-190626
  5. E. Dundar and N. Pancaroglu, Wijsman lacunary ideal invariant convergence of double sequences of sets, Honam Math. J. 42 (2020), 345-358. https://doi.org/10.5831/HMJ.2020.42.2.345
  6. A. Esi and E. Savas, On lacunary statistically convergent triple sequences in probabilistic normed space, Appl. Math. Inf. Sci. 9 (2015), 2529-2534.
  7. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
  8. A.R. Freedman and J.J. Sember, Densities and summability, Pacific J. Math. 95 (1981), 293-305. https://doi.org/10.2140/pjm.1981.95.293
  9. J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  10. M. Gurdal and M.B. Huban, $\mathcal{I}$-limit points in random 2-normed spaces, Theory Appl. Math. Comput. Sci. 2 (2012), 15-22.
  11. 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.
  12. M. Gurdal and M.B. Huban, On $\mathcal{I}$-convergence of double sequences in the topology induced by random 2-norms, Mat. Vesnik 66 (2014), 73-83.
  13. M. Gurdal and A. Sahiner, Extremal $\mathcal{I}$-limit points of double sequences, Appl. Math. E-Notes 8 (2008), 131-137.
  14. O. Kisi and F. Nuray, A new convergence for sequences of sets, Abstr. Appl. Anal. 2013 (2013), 6 pages.
  15. P. Kostyrko, T. Salat and W. Wilczynski, $\mathcal{I}$-convergence, Real Anal. Exchange 26 (2000-01), 669-685. https://doi.org/10.2307/44154069
  16. P. Kostyrko, M. Macaj, T. Sal'at and M. Sleziak, $\mathcal{I}$-convergence and extremal $\mathcal{I}$-limit points, Math. Slovaca 55 (2005), 443-464.
  17. 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 (2019), 1955-1973. https://doi.org/10.1007/s13398-018-0591-z
  18. A. Nabiev, S. Pehlivan and M. Gurdal, On $\mathcal{I}$-Cauchy sequences, Taiwanese J. Math. 11 (2007), 569-576. https://doi.org/10.11650/twjm/1500404709
  19. A.A. Nabiev, E. Savas and M. Gurdal, Statistically localized sequences in metric spaces, J. Appl. Anal. Comput. 9 (2019), 739-746.
  20. A.A. Nabiev, E. Savas and M. Gurdal, $\mathcal{I}$-localized sequences in metric spaces, Facta Univ. Ser. Math. Inf. 35 (2020), 459-469.
  21. F. Nuray, R.F. Patterson and E. Dundar, Asymptotically lacunary statistical equivalence of double sequences of sets, Demonstratio Math. 49 (2016), 183-196. https://doi.org/10.1515/dema-2016-0016
  22. F. Nuray and B.E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 87-99.
  23. F. Nuray, U. Ulusu and E. Dundar, Lacunary statistical convergence of double sequences of sets, Soft Comput. 20 (2016), 2883-2888. https://doi.org/10.1007/s00500-015-1691-8
  24. N. Pancaroglu, E. Dundar and F. Nuray, Wijsman $\mathcal{I}$-invariant convergence of sequences of sets, Bull. Math. Anal. Appl. 11 (2019), 1-9.
  25. M.M. Rao and Z.D. Ren, Applications of Orlicz spaces, Marcel Dekker Inc., 2002.
  26. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
  27. E. Savas, On $\mathcal{I}$-lacunary statistical convergence of order α for sequences of sets, Filomat 29 (2015), 1223-1229. https://doi.org/10.2298/FIL1506223S
  28. E. Savas, On $\mathcal{I}$-lacunary statistical convergence of weight g for sequences of sets, Filomat 31 (2017), 5315-5322. https://doi.org/10.2298/FIL1716315S
  29. E. Savas and P. Das, A generalized statistical convergence via ideals, Appl. Math. Letters 24 (2011), 826-830. https://doi.org/10.1016/j.aml.2010.12.022
  30. E. Savas and S. Debnath, Lacunary statistically ϕ-convergence, Note Mat. 39 (2019), 111-119.
  31. R. Savas, Multiple λµ-statistically convergence via $\tilde{\phi}$-functions, Math. Meth. Appl. Sci. (2020), 1-8. https://doi.org/10.1002/(SICI)1099-1476(19960110)19:1<1::AID-MMA756>3.0.CO;2-J
  32. N. Subramanian and A. Esi, Wijsman rough lacunary statistical convergence on $\mathcal{I}$-Cesaro triple sequences, Internat. J. Anal. Appl. 16 (2018), 643-653.
  33. A. Sahiner, M. Gurdal and F.K. Duden, Triple sequences and their statistical convergence, Selcuk J. Appl. Math. 8 (2007), 49-55.
  34. A. Sahiner, M. Gurdal, S. Saltan and H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007), 1477-1484. https://doi.org/10.11650/twjm/1500404879
  35. A. Sahiner and B. C. Tripathy, Some $\mathcal{I}$-related properties of triple sequences, Selcuk J. Appl. Math. 9 (2008), 9-18.
  36. S. Tortop and E. Dundar, Wijsman $\mathcal{I}_2$-invariant convergence of double sequences of sets, J. Ineq. Special Func. 9 (2018), 90-100.
  37. U. Ulusu and E. Dundar, $\mathcal{I}$-lacunary statistical convergence of sequences of sets, Filomat 28 (2014), 1567-1574. https://doi.org/10.2298/FIL1408567U
  38. U. Ulusu and E. Dundar, Asymptotically $\mathcal{I}_2$-lacunary statistical equivalence of double sequences of sets, J. Ineq. Special Func. 7 (2016), 44-56.
  39. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets, Progress Appl. Math. 4 (2012), 99-109.
  40. U. Ulusu and F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3 (2013), 75-88.
  41. U. Yamanci and M. Gurdal, $\mathcal{I}$-statistical convergence in 2-normed space, Arab J. Math. Sci. 20 (2014), 41-47. https://doi.org/10.1016/j.ajmsc.2013.03.001
  42. R.A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70 (1964), 186-188. https://doi.org/10.1090/S0002-9904-1964-11072-7
  43. R.A. Wijsman, Convergence of Sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc. 123 (1966), 32-45. https://doi.org/10.1090/S0002-9947-1966-0196599-8