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

Nimet Pancaroğlu Akın; Erdinç Dündar

doi:10.33434/cams.1363596

EN

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

Abstract

In the study conducted here, we have given some new concepts in summability. In this sense, firstly, we have given the concept of lacunary $\mathcal{I}_2^{\ast}$-convergence and we have investigated the relations between lacunary $\mathcal{I}_2$-convergence and lacunary $\mathcal{I}_2^{\ast}$-convergence. Also, we have given the concept of lacunary $\mathcal{I}_2^{\ast}$-Cauchy sequence and investigated the relations between lacunary $\mathcal{I}_2$-Cauchy sequence and lacunary $\mathcal{I}_2^{\ast}$-Cauchy sequence.

Keywords

Double sequence, Ideal convergence, Ideal Cauchy sequence, Lacunary sequence

References

[1] H. Fast, Sur la convergence statistique, Colloq. Math. 2(1951), 241–244.
[2] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(1959), 361–375.
[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] P. Das, E. Savaş, S.Kr. Ghosal, On generalized of certain summability methods using ideals, Appl. Math. Letter, 36(2011), 1509–1514.
[6] U. Yamancı, M. Gürdal, On lacunary ideal convergence in random n-normed space, J. Math., Volume 2013, Article ID 868457, 8 pages http://dx.doi.org/10.1155/2013/868457
[7] P. Debnath, Lacunary ideal convergence in intuitionistic fuzzy normed linear spaces, Comput. Math. Appl., 63(2012), 708–715.
[8] B.C. Tripathy, B. Hazarika, B. Choudhary, Lacunary $\mathcal{I}$-convergent sequences, Kyungpook Math. J. 52(2012), 473–482.
[9] N.P. Akın, E. Dündar, Ş. Yalvaç, Lacunary I$\mathcal{I}^{\ast }$-convergence and lacunary $\mathcal{I}^{\ast }$-Cauchy sequence, AKU J. Sci. Eng (in press).
[10] 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).

[11] P. Das, P. Kostyrko, W. Wilczynski, P. Malik, $\mathcal{I}$ and $\mathcal{I}^{\ast }$-convergence of double sequences, Math. Slovaca, 58(5)(2008), 605–620.
[12] E. Dündar, B. Altay, $\mathcal{I}_2$-convergence and $\mathcal{I}_2$-Cauchy of double sequences, Acta Math. Sci., 34B(2)(2014), 343–353.
[13] 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.
[14] E. Dündar, U. Ulusu, N. Pancaroˇglu, 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.
[15] B. Hazarika, Lacunary ideal convergence of multiple sequences, J. Egyptian Math. Soc., 24(2016), 54–59.
[16] E. Dündar, U. Ulusu, B. Aydın, $\mathcal{I}_2$-lacunary statistical convergence of double sequences of sets, Konuralp J. Math., 5(1)(2017), 1–10.
[17] A.R. Freedman, J.J. Sember, M. Raphael, Some Cesaro type summability spaces, Proc. Lond. Math. Soc. 37(1978), 508–520.
[18] Y. Sever, U. Ulusu, E. Dündar, On strongly I$\mathcal{I}$and I$\mathcal{I}^{\ast }$-lacunary convergence of sequences of sets, AIP Conf. Proc., 1611(357)(2014); doi: 10.1063/1.4893860, 7 pages.
[19] U. Ulusu, F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Inform., 3(3)(2013), 75–88.
[20] U. Ulusu, E. Dündar, I-lacunary statistical convergence of sequences of sets, Filomat, 28(8)(2014), 1567–1574.

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Nimet Pancaroğlu Akın
0000-0003-4661-5388
Türkiye

Erdinç Dündar ^*
0000-0002-0545-7486
Türkiye

Early Pub Date

November 7, 2023

Publication Date

December 25, 2023

Submission Date

September 20, 2023

Acceptance Date

November 1, 2023

Published in Issue

Year 2023 Volume: 6 Number: 4

DOI

https://doi.org/10.33434/cams.1363596

IZ

https://izlik.org/JA76ZW29ZE

APA

Pancaroğlu Akın, N., & Dündar, E. (2023). On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Communications in Advanced Mathematical Sciences, 6(4), 188-195. https://doi.org/10.33434/cams.1363596

AMA

1.Pancaroğlu Akın N, Dündar E. On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Communications in Advanced Mathematical Sciences. 2023;6(4):188-195. doi:10.33434/cams.1363596

Chicago

Pancaroğlu Akın, Nimet, and Erdinç Dündar. 2023. “On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Communications in Advanced Mathematical Sciences 6 (4): 188-95. https://doi.org/10.33434/cams.1363596.

EndNote

Pancaroğlu Akın N, Dündar E (December 1, 2023) On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Communications in Advanced Mathematical Sciences 6 4 188–195.

IEEE

[1]N. Pancaroğlu Akın and E. Dündar, “On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”, Communications in Advanced Mathematical Sciences, vol. 6, no. 4, pp. 188–195, Dec. 2023, doi: 10.33434/cams.1363596.

ISNAD

Pancaroğlu Akın, Nimet - Dündar, Erdinç. “On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Communications in Advanced Mathematical Sciences 6/4 (December 1, 2023): 188-195. https://doi.org/10.33434/cams.1363596.

JAMA

1.Pancaroğlu Akın N, Dündar E. On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Communications in Advanced Mathematical Sciences. 2023;6:188–195.

MLA

Pancaroğlu Akın, Nimet, and Erdinç Dündar. “On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence”. Communications in Advanced Mathematical Sciences, vol. 6, no. 4, Dec. 2023, pp. 188-95, doi:10.33434/cams.1363596.

Vancouver

1.Nimet Pancaroğlu Akın, Erdinç Dündar. On Lacunary $\mathcal{I}_2^{\ast }$-Convergence and Lacunary $\mathcal{I}_2^{\ast }$-Cauchy Sequence. Communications in Advanced Mathematical Sciences. 2023 Dec. 1;6(4):188-95. doi:10.33434/cams.1363596

Cited By

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

Universal Journal of Mathematics and Applications

https://doi.org/10.32323/ujma.1376849

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