Research Article
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Year 2023, , 155 - 161, 18.12.2023
https://doi.org/10.32323/ujma.1376849

Abstract

References

  • [1] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
  • [2] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • [3] P. Kostyrko, T. Salat, W. Wilczynski, $\mathcal{I}$ -Convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
  • [4] A. Nabiev, S. Pehlivan, M. Gürdal, On $\mathcal{I}$ -Cauchy sequence, Taiwanese J. Math., 11(2) (2007), 569–576.
  • [5] U. Yamancı, M. Gürdal, On lacunary ideal convergence in random n-normed space, J. Math., 2013 (2013), Article ID 868457, 8 pages.
  • [6] B. C. Tripathy, B. Hazarika, B. Choudhary, Lacunary $\mathcal{I}$ -convergent sequences, Kyungpook Math. J., 52 (2012), 473–482.
  • [7] N. P. Akın, E. Dündar, S¸ . Yalvaç, Lacunary $\mathcal{I}^{\ast }$ -convergence and lacunary $\mathcal{I}^{\ast }$ -Cauchy sequence, AKU J. Sci. Eng., (in press).
  • [8] N. P. Akın, E. Dündar, Strongly lacunary $\mathcal{I}^{\ast }$ -convergence and strongly lacunary $\mathcal{I}^{\ast }$ -Cauchy sequence, Math. Sci. Appl. E-Notes, (in press).
  • [9] P. Das, P. Kostyrko, W. Wilczynski, P. Malik, $\mathcal{I}$ and $\mathcal{I}^*$ -convergence of double sequences, Math. Slovaca, 58(5) (2008), 605-620.
  • [10] E. Dündar, B. Altay, $\mathcal{I}_2$ -convergence and $\mathcal{I}_2$ -Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343–353.
  • [11] E. Dündar, B. Altay, On some properties of $\mathcal{I}_2$ -convergence and $\mathcal{I}_2$ -Cauchy of double sequences, Gen. Math. Notes, 7(1) (2011) 1–12.
  • [12] B. Hazarika, Lacunary ideal convergence of multiple sequences, J. Egyptian Math. Soc., 24 (2016), 54–59.
  • [13] E. Dündar, U. Ulusu, N. Pancaroğlu, Strongly $\mathcal{I}_2$ -lacunary convergence and $\mathcal{I}_2$ -lacunary Cauchy double sequences of sets, Aligarh Bull. Math., 35(1-2) (2016), 1–15.
  • [14] N. P. Akın, E. Dündar, On lacunary $\mathcal{I}_2^{\ast }$ -convergence and lacunary $\mathcal{I}_2^{\ast }$ -Cauchy sequence, Commun. Adv. Math. Sci., 6(4) (2023), 188–195.
  • [15] P. Das, E. Savaş, S. Kr. Ghosal, On generalized of certain summability methods using ideals, Appl. Math. Letter, 36 (2011), 1509–1514.
  • [16] P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (2012), 708–715.
  • [17] E. Dündar, U. Ulusu, On rough $\mathcal{I}$ -convergence and $\mathcal{I}$ -Cauchy sequence for functions defined on amenable semigroup, Univer. J. Math. Appl., 6(2) (2023), 86–90.
  • [18] A. R. Freedman, J. J. Sember, M. Raphael, Some Cesaro type summability spaces, Proc. Lond. Math. Soc., 37 (1978), 508–520.
  • [19] F. Nuray, E. Dündar, U. Ulusu, Wijsman $\mathcal{I}_2$ -convergence of double sequences of closed sets, Pure Appl. Math. Lett., 2 (2014), 35–39.
  • [20] Y. Sever, U. Ulusu, E. Dündar, On strongly $\mathcal{I}$ and $\mathcal{I}*$ -lacunary convergence of sequences of sets, AIP Conf. Proc., 1611 (2014), 357–362.
  • [21] U. Ulusu, F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3(3) (2013), 75–88.

On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence

Year 2023, , 155 - 161, 18.12.2023
https://doi.org/10.32323/ujma.1376849

Abstract

In the study conducted here, we have given some new concepts in summability theory. In this sense, firstly, using the lacunary sequence we have given the concept of strongly $\mathcal{I}_{\theta_2}^{\ast}$-convergence and we have examined the relations between $\mathcal{I}_{\theta_2}^{\ast}$-convergence and strongly $\mathcal{I}_{\theta_2}^{\ast}$-convergence and also between strongly $\mathcal{I}_{\theta_2}$-convergence and strongly $\mathcal{I}_{\theta_2}^{\ast}$-convergence. Also, using the lacunary sequence we have given the concept of strongly $\mathcal{I}_{\theta_2}^{\ast}$-Cauchy sequence and examined the relations between strongly $\mathcal{I}_{\theta_2}$-Cauchy sequence and strongly $\mathcal{I}_{\theta_2}^{\ast}$-Cauchy sequence.

References

  • [1] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
  • [2] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
  • [3] P. Kostyrko, T. Salat, W. Wilczynski, $\mathcal{I}$ -Convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
  • [4] A. Nabiev, S. Pehlivan, M. Gürdal, On $\mathcal{I}$ -Cauchy sequence, Taiwanese J. Math., 11(2) (2007), 569–576.
  • [5] U. Yamancı, M. Gürdal, On lacunary ideal convergence in random n-normed space, J. Math., 2013 (2013), Article ID 868457, 8 pages.
  • [6] B. C. Tripathy, B. Hazarika, B. Choudhary, Lacunary $\mathcal{I}$ -convergent sequences, Kyungpook Math. J., 52 (2012), 473–482.
  • [7] N. P. Akın, E. Dündar, S¸ . Yalvaç, Lacunary $\mathcal{I}^{\ast }$ -convergence and lacunary $\mathcal{I}^{\ast }$ -Cauchy sequence, AKU J. Sci. Eng., (in press).
  • [8] N. P. Akın, E. Dündar, Strongly lacunary $\mathcal{I}^{\ast }$ -convergence and strongly lacunary $\mathcal{I}^{\ast }$ -Cauchy sequence, Math. Sci. Appl. E-Notes, (in press).
  • [9] P. Das, P. Kostyrko, W. Wilczynski, P. Malik, $\mathcal{I}$ and $\mathcal{I}^*$ -convergence of double sequences, Math. Slovaca, 58(5) (2008), 605-620.
  • [10] E. Dündar, B. Altay, $\mathcal{I}_2$ -convergence and $\mathcal{I}_2$ -Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343–353.
  • [11] E. Dündar, B. Altay, On some properties of $\mathcal{I}_2$ -convergence and $\mathcal{I}_2$ -Cauchy of double sequences, Gen. Math. Notes, 7(1) (2011) 1–12.
  • [12] B. Hazarika, Lacunary ideal convergence of multiple sequences, J. Egyptian Math. Soc., 24 (2016), 54–59.
  • [13] E. Dündar, U. Ulusu, N. Pancaroğlu, Strongly $\mathcal{I}_2$ -lacunary convergence and $\mathcal{I}_2$ -lacunary Cauchy double sequences of sets, Aligarh Bull. Math., 35(1-2) (2016), 1–15.
  • [14] N. P. Akın, E. Dündar, On lacunary $\mathcal{I}_2^{\ast }$ -convergence and lacunary $\mathcal{I}_2^{\ast }$ -Cauchy sequence, Commun. Adv. Math. Sci., 6(4) (2023), 188–195.
  • [15] P. Das, E. Savaş, S. Kr. Ghosal, On generalized of certain summability methods using ideals, Appl. Math. Letter, 36 (2011), 1509–1514.
  • [16] P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63 (2012), 708–715.
  • [17] E. Dündar, U. Ulusu, On rough $\mathcal{I}$ -convergence and $\mathcal{I}$ -Cauchy sequence for functions defined on amenable semigroup, Univer. J. Math. Appl., 6(2) (2023), 86–90.
  • [18] A. R. Freedman, J. J. Sember, M. Raphael, Some Cesaro type summability spaces, Proc. Lond. Math. Soc., 37 (1978), 508–520.
  • [19] F. Nuray, E. Dündar, U. Ulusu, Wijsman $\mathcal{I}_2$ -convergence of double sequences of closed sets, Pure Appl. Math. Lett., 2 (2014), 35–39.
  • [20] Y. Sever, U. Ulusu, E. Dündar, On strongly $\mathcal{I}$ and $\mathcal{I}*$ -lacunary convergence of sequences of sets, AIP Conf. Proc., 1611 (2014), 357–362.
  • [21] U. Ulusu, F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3(3) (2013), 75–88.
There are 21 citations in total.

Details

Primary Language English
Subjects Pure Mathematics (Other)
Journal Section Articles
Authors

Erdinç Dündar 0000-0002-0545-7486

Nimet Pancaroğlu Akın 0000-0003-4661-5388

Esra Gülle 0000-0001-5575-2937

Early Pub Date December 4, 2023
Publication Date December 18, 2023
Submission Date October 16, 2023
Acceptance Date November 30, 2023
Published in Issue Year 2023

Cite

APA Dündar, E., Pancaroğlu Akın, N., & Gülle, E. (2023). On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Universal Journal of Mathematics and Applications, 6(4), 155-161. https://doi.org/10.32323/ujma.1376849
AMA Dündar E, Pancaroğlu Akın N, Gülle E. On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Univ. J. Math. Appl. December 2023;6(4):155-161. doi:10.32323/ujma.1376849
Chicago Dündar, Erdinç, Nimet Pancaroğlu Akın, and Esra Gülle. “On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Universal Journal of Mathematics and Applications 6, no. 4 (December 2023): 155-61. https://doi.org/10.32323/ujma.1376849.
EndNote Dündar E, Pancaroğlu Akın N, Gülle E (December 1, 2023) On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Universal Journal of Mathematics and Applications 6 4 155–161.
IEEE E. Dündar, N. Pancaroğlu Akın, and E. Gülle, “On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”, Univ. J. Math. Appl., vol. 6, no. 4, pp. 155–161, 2023, doi: 10.32323/ujma.1376849.
ISNAD Dündar, Erdinç et al. “On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Universal Journal of Mathematics and Applications 6/4 (December 2023), 155-161. https://doi.org/10.32323/ujma.1376849.
JAMA Dündar E, Pancaroğlu Akın N, Gülle E. On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Univ. J. Math. Appl. 2023;6:155–161.
MLA Dündar, Erdinç et al. “On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Universal Journal of Mathematics and Applications, vol. 6, no. 4, 2023, pp. 155-61, doi:10.32323/ujma.1376849.
Vancouver Dündar E, Pancaroğlu Akın N, Gülle E. On Strongly Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Strongly Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Univ. J. Math. Appl. 2023;6(4):155-61.

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