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

Year 2023, Volume: 6 Issue: 4, 188 - 195, 25.12.2023

Nimet Pancaroğlu Akın , Erdinç Dündar

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

Cited By: 1

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.