Journal of applied mathematics & informatics

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

DOI QR Code

DOUBLE WIJSMAN LACUNARY STATISTICAL CONVERGENCE OF ORDER 𝛼

  • GULLE, ESRA (Department of Mathematics, Afyon Kocatepe University) ;
  • ULUSU, UGUR (Sivas Cumhuriyet University)
  • Received : 2020.08.25
  • Accepted : 2021.03.30
  • Published : 2021.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the concepts of Wijsman strongly p-lacunary summability of order 𝛼, Wijsman lacunary statistical convergence of order 𝛼 and Hausdorff lacunary statistical convergence of order 𝛼 for double set sequences. Also, we investigate some properties of these new concepts and examine the existence of some relationships between them. Furthermore, we study the relationships between these new concepts and some concepts in the literature.

Keywords

References

  1. M. Baronti and P. Papini, Convergence of sequences of sets, Methods of Functional Analysis in Approximation Theory 76 (1986), 133-155.
  2. G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Aust. Math. Soc. 31 (1985), 421-432. https://doi.org/10.1017/S0004972700009370
  3. G. Beer, Wijsman convergence:A survey, Set-Valued Var. Anal. 2 (1994), 77-94. https://doi.org/10.1007/BF01027094
  4. S. Bhunia, P. Das and S.K. Pal, Restricting statistical convergence, Acta Math. Hungar. 134 (2012), 153-161. https://doi.org/10.1007/s10474-011-0122-2
  5. J.S. 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
  6. H. Cakalli and E. Savas, Statistical convergence of double sequences in topological groups, J. Comput. Anal. Appl. 12 (2010), 421-426.
  7. R. Colak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications 1 (2010), 121-129.
  8. R. Colak, C.A. Bektas, λ-statistical convergence of order α, Acta Math. Sci. Ser. B 31 (2011), 953-959. https://doi.org/10.1016/S0252-9602(11)60288-9
  9. R. Colak and Y. Altin, Statistical convergence of double sequences of order α, J. Funct. Spaces Appl. 2013 (2013), 1-5.
  10. E. Dundar and B. Altay, Multipliers for bounded $\mathcal{I}_2$-convergent of double sequences, Math. Comput. Modelling 55 (2012), 1193-1198. https://doi.org/10.1016/j.mcm.2011.09.043
  11. E. Dundar and N. Pancaroglu Akin, Wijsman regularly ideal convergence of double sequences of sets, J. Intell. Fuzzy Systems 37 (2019), 8159-8166, https://doi.org/10.3233/JIFS-190626
  12. M. Et and H. Sengul, Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat 28 (2014), 1593-1602. https://doi.org/10.2298/FIL1408593E
  13. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244. https://doi.org/10.4064/cm-2-3-4-241-244
  14. A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc. Lond. Math. Soc. 37 (1978), 508-520.
  15. J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301-313. https://doi.org/10.1524/anly.1985.5.4.301
  16. J.A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993), 43-52. https://doi.org/10.2140/pjm.1993.160.43
  17. A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2001), 129-138. https://doi.org/10.1216/rmjm/1030539612
  18. F. Moricz, Statistical convergence of multiple sequences, Arch. Math. 81 (2003), 82-89. https://doi.org/10.1007/s00013-003-0506-9
  19. 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
  20. M. Mursaleen, C. Cakan, S.A. Mohiuddine and E. Savas, Generalized statistical convergence and statistical core of double sequences, Acta Math. Sin. 26 (2010), 2131-2144. https://doi.org/10.1007/s10114-010-9050-2
  21. F. Nuray and B.E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 87-99.
  22. F. Nuray, U. Ulusu and E. Dundar, Cesaro summability of double sequences of sets, Gen. Math. Notes 25 (2014), 8-18.
  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. F. Nuray, E. Dundar and U. Ulusu, Wijsman statistical convergence of double sequences of sets, Iran. J. Math. Sci. Inform. 16 (2021), 55-64.
  25. R.F. Patterson and E. Savas, Lacunary statistical convergence of double sequences, Math. Commun. 10 (2005), 55-61.
  26. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321. https://doi.org/10.1007/BF01448977
  27. E. Savas, Some sequence spaces and statistical convergence, Int. J. Math. Math. Sci. 29 (2002), 303-306. https://doi.org/10.1155/S0161171202003071
  28. E. Savas and R.F. Patterson, Lacunary statistical convergence of multiple sequences, Appl. Math. Lett. 19 (2006), 527-534. https://doi.org/10.1016/j.aml.2005.06.018
  29. E. Savas, Double almost statistical convergence of order α, Adv. Difference Equ. 2013 (2013), 1-9. https://doi.org/10.1186/1687-1847-2013-1
  30. E. Savas, Double almost lacunary statistical convergence of order α, Adv. Difference Equ. 2013 (2013), 1-10. https://doi.org/10.1186/1687-1847-2013-1
  31. 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
  32. I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375. https://doi.org/10.2307/2308747
  33. Y. Sever, O. Talo and B. Altay, On convergence of double sequences of closed sets, Contemporary Analysis and Applied Mathematics 3 (2015), 30-49.
  34. 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
  35. T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139-150.
  36. H. Sengul and M. Et, On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B 34 (2014), 473-482. https://doi.org/10.1016/S0252-9602(14)60021-7
  37. H. Sengul and M. Et, On $\mathcal{I}$-lacunary statistical convergence of order α of sequences of sets, Filomat 31 (2017), 2403-2412. https://doi.org/10.2298/FIL1708403S
  38. O. Talo, Y. Sever and F. Basar, On statistically convergent sequences of closed sets, Filomat 30 (2016), 1497-1509. https://doi.org/10.2298/FIL1606497T
  39. S. Tortop and E. Dundar, Wijsman $\mathcal{I}_2$-invariant convergence of double sequences of sets, Journal of Inequalities and Special Functions 9 (2018), 90-100.
  40. B.C. Tripathy, On statistical convergence, Proc. Est. Acad. Sci. 47 (1998), 299-303.
  41. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Progress in Applied Mathematics 4 (2012), 99-109.
  42. U. Ulusu and E. Gulle, Some statistical convergence types for double set sequences of order α, Facta Universitatis Ser. Math. Inform. 35 (2020), 595-603. https://doi.org/10.22190/FUMI2003595U
  43. 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