$\mathcal{I}_2$-Convergence of Double Sequences of Functions in 2-Normed Spaces
Year 2019,
Volume: 2 Issue: 3, 130 - 137, 30.09.2019
Sevim Yegül
Erdinç Dündar
Abstract
In this study, we introduced the concepts of $\mathcal{I}_2$-convergence and $\mathcal{I}_2^*$-convergence of double sequences of functions in $2$-normed space. Also, were studied some properties about these concepts and investigated relationships between them for double sequences of functions in $2$-normed spaces.
References
- [1] M. Arslan, E. Dündar, I-Convergence and I-Cauchy Sequence of Functions In 2-Normed Spaces, Konuralp J. Math., 6(1) (2018), 57–62.
- [2] M. Arslan, E. Dündar, On I-Convergence of sequences of functions in 2-normed spaces, Southeast Asian Bull. Math., 42 (2018), 491–502.
- [3] M. Arslan, E. Dündar, Rough convergence in 2-normed spaces, Bull. Math. Anal. Appl., 10(3) (2018), 1–9.
- [4] V. Balaz, J. Cervenansky, P. Kostyrko, T. Salat, I-convergence and I-continuity of real functions, Acta Math., Faculty of Natural Sciences, Constantine
the Philosopher University, Nitra, 5 (2004), 43–50.
- [5] H. Çakalli, S. Ersan, New types of continuity in 2-normed spaces, Filomat, 30(3) (2016), 525–532.
- [6] P. Das, P. Kostyrko, W. Wilczy´nski, P. Malik, I and I-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605–620.
- [7] E. Dündar, B. Altay, I2-convergence of double sequences of functions, Electron. J. Math. Anal. Appl., 3(1) (2015), 111–121.
- [8] E. Dündar, B. Altay, I2-convergence and I2-Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343–353.
- [9] E. Dündar, B. Altay, I2-uniform convergence of double sequences of functions Filomat, 30(5) (2016), 1273–1281.
- [10] E. Dündar, B. Altay, Multipliers for bounded I2-convergent of double sequences, Math. Comput. Modelling, 55(3-4) (2012), 1193–1198.
- [11] E. Dündar, On some results of I2-convergence of double sequences of functions, Math. Anal. Sci. Appl. E-notes, 3(1) (2015), 44–52.
- [12] E. Dündar. Ö. Talo, I2-convergence of double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 10(3) (2013), 37–50.
- [13] E. Dündar, M. Arslan, S. Yegül, On I-Uniform Convergence of Sequences of Functions In 2-Normed Spaces, (Under Review).
- [14] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
- [15] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
- [16] S. Gahler, 2-metrische R¨aume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148.
- [17] S. Gahler, 2-normed spaces, Math. Nachr. 28 (1964), 1–43.
- [18] F. Gezer, S.Karakus¸, I and I convergent function sequences, Math. Commun. 10 (2005), 71–80.
- [19] A. Gökhan, M. Güngör, M. Et, Statistical convergence of double sequences of real-valued functions, Int. Math. Forum, 2(8) (2007), 365–374.
- [20] H. Gunawan, M. Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math. 27(3) (2001), 321–329.
- [21] M. Gürdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2(1) (2004), 107–113.
- [22] M. Gürdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math., 33 (2009), 257–264.
- [23] M. Gürdal, I.Açık On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11(2) (2008), 349–354.
- [24] M. Gürdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4(1) (2006), 85–91.
- [25] P. Kostyrko, T. Salat, W. Wilczynski, I-convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
- [26] M. Mursaleen, A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca, 61(6) (2011), 933–940.
- [27] N. Pancaroğlu, E. Dündar, F. Nuray, Wijsman I-Invariant Convergence of Sequences of Sets, Bull. Math. Anal. Appl., (Accepted - in press).
- [28] N. Pancaroğlu, E. Dündar, U. Ulusu, Wijsman Lacunary I-Invariant Convergence of Sequences of Sets, (Under Review).
- [29] S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces, Internat. J. Math. Math. Sci. 2011
(2011), 10 pages.
- [30] A. S¸ ahiner, M. Gürdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007), 1477–1484.
- [31] E. Savaş, M. Gürdal, Ideal Convergent Function Sequences in Random 2-Normed Spaces, Filomat, 30(3) (2016), 557–567.
- [32] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
- [33] Y. Sever, E. Dündar, Regularly ideal convergence and regularly ideal Cauchy double sequences in 2-normed spaces, Filomat, 28(5) (2015), 907–915.
- [34] Ş . Tortop, E. Dündar, Wijsman I2 invariant convergence of Double Sequences of Sets, 9(4) (2018), 90–100.
- [35] U. Ulusu, E. D¨undar, F. Nuray, Lacunary I2-Invariant Convergence and Some Properties, Internat. J. Anal. Appl., 16(3) (2018), 317–327.
- [36] U. Ulusu, E. D¨undar, I-Lacunary Statistical Convergence of Sequences of Sets, Filomat, 28(8) (2013), 1567–1574.
- [37] M. R. T¨urkmen and M. Çınar, Lambda Statistical Convergence in Fuzzy Normed Linear Spaces, J. Intel. Fuzzy Sys., 34(6) (2018), 4023–4030
- [38] M. R. Türkmen and E. Dündar, On Lacunary Statistical Convergence of Double Sequences and Some Properties in Fuzzy Normed Spaces, J. Intel.
Fuzzy Sys., DOI: 10.3233/JIFS-18841 (Pre-press).
- [39] S. Yegül, E. Dündar, On Statistical Convergence of Sequences of Functions In 2-Normed Spaces, J. Class. Anal., (2017); 10(1):49–57.
- [40] S. Yegül, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Univ. Ser. Math.
Inform., 33(5) (2018), 705–719.
Year 2019,
Volume: 2 Issue: 3, 130 - 137, 30.09.2019
Sevim Yegül
Erdinç Dündar
References
- [1] M. Arslan, E. Dündar, I-Convergence and I-Cauchy Sequence of Functions In 2-Normed Spaces, Konuralp J. Math., 6(1) (2018), 57–62.
- [2] M. Arslan, E. Dündar, On I-Convergence of sequences of functions in 2-normed spaces, Southeast Asian Bull. Math., 42 (2018), 491–502.
- [3] M. Arslan, E. Dündar, Rough convergence in 2-normed spaces, Bull. Math. Anal. Appl., 10(3) (2018), 1–9.
- [4] V. Balaz, J. Cervenansky, P. Kostyrko, T. Salat, I-convergence and I-continuity of real functions, Acta Math., Faculty of Natural Sciences, Constantine
the Philosopher University, Nitra, 5 (2004), 43–50.
- [5] H. Çakalli, S. Ersan, New types of continuity in 2-normed spaces, Filomat, 30(3) (2016), 525–532.
- [6] P. Das, P. Kostyrko, W. Wilczy´nski, P. Malik, I and I-convergence of double sequences, Math. Slovaca, 58(5) (2008), 605–620.
- [7] E. Dündar, B. Altay, I2-convergence of double sequences of functions, Electron. J. Math. Anal. Appl., 3(1) (2015), 111–121.
- [8] E. Dündar, B. Altay, I2-convergence and I2-Cauchy of double sequences, Acta Math. Sci., 34B(2) (2014), 343–353.
- [9] E. Dündar, B. Altay, I2-uniform convergence of double sequences of functions Filomat, 30(5) (2016), 1273–1281.
- [10] E. Dündar, B. Altay, Multipliers for bounded I2-convergent of double sequences, Math. Comput. Modelling, 55(3-4) (2012), 1193–1198.
- [11] E. Dündar, On some results of I2-convergence of double sequences of functions, Math. Anal. Sci. Appl. E-notes, 3(1) (2015), 44–52.
- [12] E. Dündar. Ö. Talo, I2-convergence of double sequences of fuzzy numbers, Iran. J. Fuzzy Syst., 10(3) (2013), 37–50.
- [13] E. Dündar, M. Arslan, S. Yegül, On I-Uniform Convergence of Sequences of Functions In 2-Normed Spaces, (Under Review).
- [14] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
- [15] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
- [16] S. Gahler, 2-metrische R¨aume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148.
- [17] S. Gahler, 2-normed spaces, Math. Nachr. 28 (1964), 1–43.
- [18] F. Gezer, S.Karakus¸, I and I convergent function sequences, Math. Commun. 10 (2005), 71–80.
- [19] A. Gökhan, M. Güngör, M. Et, Statistical convergence of double sequences of real-valued functions, Int. Math. Forum, 2(8) (2007), 365–374.
- [20] H. Gunawan, M. Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math. 27(3) (2001), 321–329.
- [21] M. Gürdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2(1) (2004), 107–113.
- [22] M. Gürdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math., 33 (2009), 257–264.
- [23] M. Gürdal, I.Açık On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11(2) (2008), 349–354.
- [24] M. Gürdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4(1) (2006), 85–91.
- [25] P. Kostyrko, T. Salat, W. Wilczynski, I-convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
- [26] M. Mursaleen, A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca, 61(6) (2011), 933–940.
- [27] N. Pancaroğlu, E. Dündar, F. Nuray, Wijsman I-Invariant Convergence of Sequences of Sets, Bull. Math. Anal. Appl., (Accepted - in press).
- [28] N. Pancaroğlu, E. Dündar, U. Ulusu, Wijsman Lacunary I-Invariant Convergence of Sequences of Sets, (Under Review).
- [29] S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces, Internat. J. Math. Math. Sci. 2011
(2011), 10 pages.
- [30] A. S¸ ahiner, M. Gürdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11 (2007), 1477–1484.
- [31] E. Savaş, M. Gürdal, Ideal Convergent Function Sequences in Random 2-Normed Spaces, Filomat, 30(3) (2016), 557–567.
- [32] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
- [33] Y. Sever, E. Dündar, Regularly ideal convergence and regularly ideal Cauchy double sequences in 2-normed spaces, Filomat, 28(5) (2015), 907–915.
- [34] Ş . Tortop, E. Dündar, Wijsman I2 invariant convergence of Double Sequences of Sets, 9(4) (2018), 90–100.
- [35] U. Ulusu, E. D¨undar, F. Nuray, Lacunary I2-Invariant Convergence and Some Properties, Internat. J. Anal. Appl., 16(3) (2018), 317–327.
- [36] U. Ulusu, E. D¨undar, I-Lacunary Statistical Convergence of Sequences of Sets, Filomat, 28(8) (2013), 1567–1574.
- [37] M. R. T¨urkmen and M. Çınar, Lambda Statistical Convergence in Fuzzy Normed Linear Spaces, J. Intel. Fuzzy Sys., 34(6) (2018), 4023–4030
- [38] M. R. Türkmen and E. Dündar, On Lacunary Statistical Convergence of Double Sequences and Some Properties in Fuzzy Normed Spaces, J. Intel.
Fuzzy Sys., DOI: 10.3233/JIFS-18841 (Pre-press).
- [39] S. Yegül, E. Dündar, On Statistical Convergence of Sequences of Functions In 2-Normed Spaces, J. Class. Anal., (2017); 10(1):49–57.
- [40] S. Yegül, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Univ. Ser. Math.
Inform., 33(5) (2018), 705–719.