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

Year 2023, , 155 - 161, 18.12.2023

Erdinç Dündar , Nimet Pancaroğlu Akın , Esra Gülle

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

Cited By: 1

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.

Keywords

$\mathcal{I}_2$-Cauchy Sequence, $\mathcal{I}_2$-Convergence, Double sequence, Ideal, Lacunary 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.