Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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WIJSMAN ASYMPTOTICAL ${\mathcal{I}}_2$-LACUNARY STATISTICAL EQUIVALENCE OF ORDER 𝜂 FOR DOUBLE SET SEQUENCES

  • GULLE, ESRA (Department of Mathematics, Afyon Kocatepe University) ;
  • ULUSU, UGUR (Sivas Cumhuriyet University)
  • Received : 2020.12.08
  • Accepted : 2021.03.16
  • Published : 2022.01.30
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Abstract

In this paper, for double set sequences, as a new approach to the notion of Wijsman asymptotical lacunary statistical equivalence of order 𝜂, we introduce new concepts which are called Wijsman asymptotical ${\mathcal{I}}_2$-lacunary statistical equivalence of order 𝜂 and Wijsman asymptotical strong ${\mathcal{I}}_2$-lacunary equivalence of order 𝜂 where 0 < 𝜂 ≤ 1. Also, some properties of these new concepts are investigated, and the existence of some relations between these and some previously studied asymptotical equivalence concepts for double set sequences is examined.

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References

  1. Y. Altin, R. Colak and B. Torgut, ${\mathcal{I}}_2(u)$-convergence of double sequences of order (α, β), Georgian Mathematical Journal 22 (2015), 153-158.
  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 Anal. 2 (1994), 77-94. https://doi.org/10.1007/BF01027094
  4. S. Bhunia, P. Das and S.K. Pal, Restricting statistical convergence, Acta Mathematica Hungarica 134 (2012), 153-161. https://doi.org/10.1007/s10474-011-0122-2
  5. R. Colak, Statistical convergence of order α, Mordern Methods in Analysis and Its Applications, Anamaya Publishers, New Delhi, 2010.
  6. R. Colak and Y. Altin, Statistical convergence of double sequences of order α, Journal of Function Spaces and Applications 2013 (2013), 5 pages.
  7. P. Das, P. Kostyrko, W. Wilczynski and P. Malik, $\mathcal{I}$ and $\mathcal{I}$*-convergence of double sequences, Math. Slovaca 58 (2008), 605-620. https://doi.org/10.2478/s12175-008-0096-x
  8. P. Das and E. Savas, On $mathcal{I}$-statistical and $mathcal{I}$-lacunary statistical convergence of order α, Bull. Iranian Math. Soc. 40 (2014), 459-472.
  9. E. Dundar, U. Ulusu and B. Aydin, $mathcal{I}_2$-lacunary statistical convergence of double sequences of sets, Konuralp Journal of Mathematics 5 (2017), 1-10.
  10. A. Esi, On asymptotically double lacunary statistically equivalent sequences, Applied Mathematics Letters 22 (2009), 1781-1785. https://doi.org/10.1016/j.aml.2009.06.018
  11. A. Esi and M. Acikgoz, On λ2-asymptotically double statistical equivalent sequences, Int. J. Nonlinear Anal. Appl. 5 (2014), 16-21.
  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. A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, The Rocky Mountain Journal of Mathematics 32 (2001), 129-138. https://doi.org/10.1216/rmjm/1030539612
  14. E. Gulle, Double Wijsman asymptotically statistical equivalence of order α, Journal of Intelligent and Fuzzy Systems 38 (2020), 2081-2087. https://doi.org/10.3233/jifs-190796
  15. B. Hazarika and V. Kumar, On asymptotically double lacunary statistical equivalent sequences in ideal context, Journal of Inequalities and Applications 2013 (2013), 15 pages.
  16. O. Kisi and F. Nuray, On Sλ($\mathcal{I}$)-asymptotically statistical equivalence of sequence of sets, ISRN Mathematical Analysis 2013 (2013), 6 pages.
  17. P. Kostyrko, T. Salat and W. Wilczynski, $mathcal{I}$-convergence, Real Anal. Exchange 26 (2000), 669-686. https://doi.org/10.2307/44154069
  18. 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
  19. M. Mursaleen, S.A. Mohiuddine and O.H.H. Edely, On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Comput. Math. Appl. 59 (2010), 603-611. https://doi.org/10.1016/j.camwa.2009.11.002
  20. M. Mursaleen and S.A. Mohiuddine, On ideal convergence of double sequences in probabilistic normed spaces, Math. Reports 12 (2010), 359-371.
  21. M. Mursaleen and A. Alotaibi, On $mathcal{I}$-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 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
  23. M. Mursaleen, S. Debnath and D. Rakshit, $mathcal{I}$-statistical limit superior and $mathcal{I}$-statistical limit inferior, Filomat 31 (2017), 2103-2108. https://doi.org/10.2298/FIL1707103M
  24. F. Nuray, E. Dundar and U. Ulusu, Wijsman $mathcal{I}_2$-convergence of double sequences of closed sets, Pure and Applied Mathematics Letters 2 (2014), 35-39.
  25. F. Nuray, U. Ulusu and E. Dundar, Lacunary statistical convergence of double sequences of sets, Soft Computing 20 (2016), 2883-2888. https://doi.org/10.1007/s00500-015-1691-8
  26. F. Nuray, R.F. Patterson and E. Dundar, Asymptotically lacunary statistical equivalence of double sequences of sets, Demonstratio Mathematica 49 (2016), 183-196. https://doi.org/10.1515/dema-2016-0016
  27. F. Nuray, E. Dundar and U. Ulusu, Wijsman statistical convergence of double sequences of sets, Iran. J. Math. Sci. Inform. 16 (2021), 55-64.
  28. R.F. Patterson, Rates of convergence for double sequences, Southeast Asian Bull. Math. 26 (2002), 469-478. https://doi.org/10.1007/s10012-002-0469-y
  29. R.F. Patterson and E. Savas, Lacunary statistical convergence of double sequences, Math. Commun. 10 (2005), 55-61.
  30. R.F. Patterson and E. Savas, On asymptotically lacunary statistical equivalent sequences, Thai J. Math. 4 (2006), 267-272.
  31. A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), 289-321. https://doi.org/10.1007/BF01448977
  32. E. Savas, On $mathcal{I}$-asymptotically lacunary statistical equivalent sequences, Advances in Difference Equations 2013 (2013), 1-7. https://doi.org/10.1186/1687-1847-2013-1
  33. E. Savas, Double almost lacunary statistical convergence of order α, Advances in Difference Equations 2013 (2013), 1-10. https://doi.org/10.1186/1687-1847-2013-1
  34. 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
  35. E. Savas, Asymptotically $mathcal{I}$-lacunary statistical equivalent of order α for sequences of sets, Journal of Nonlinear Sciences and Applications 10 (2017), 2860-2867. https://doi.org/10.22436/jnsa.010.06.01
  36. 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
  37. U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Progress in Applied Mathematics 4 (2012), 99-109.
  38. U. Ulusu and F. Nuray, On asymptotically lacunary statistical equivalent set sequences, Journal of Mathematics 2013 (2013), 5 pages.
  39. 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
  40. U. Ulusu and E. Dundar, Asymptotically $mathcal{I}_2$-lacunary statistical equivalence of double sequences of sets, Journal of Inequalities and Special Functions 7 (2016), 44-56.
  41. U. Ulusu, E. Dundar and B. Aydin, Asymptotically $mathcal{I}_2$-Cesaro equivalence of double sequences of sets, Journal of Inequalities and Special Functions 7 (2016), 225-234.
  42. U. Ulusu, E. Dundar and E. Gulle, $mathcal{I}_2$-Cesaro summability of double sequences of sets, Palestine Journal of Mathematics 9 (2020), 561-568.
  43. U. Ulusu and E. Gulle, Some statistical convergence types for double set sequences of order α, Facta Univ. Ser. Math. Inform. 35 (2020), 595-603.
  44. U. Ulusu and E. Gulle, Wijsman asymptotically $mathcal{I}_2$-statistical equivalence of order α, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (2020), 854-862. https://doi.org/10.31801/cfsuasmas.695309
  45. 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