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$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces

Year 2022, Volume: 5 Issue: 3, 150 - 160, 30.09.2022

Sevim Yegül Güzey , Erdinç Dündar , Mukaddes Arslan

https://doi.org/10.33434/cams.1177174

Abstract

In this work, we discuss various types of $\mathcal{I}_2$-uniform convergence and equi-continuous for double sequences of functions. Also, we introduce the concepts of $\mathcal{I}_2$-uniform convergence, $\mathcal{I}_2^*$-uniform convergence, $\mathcal{I}_2$-uniformly Cauchy sequences and $\mathcal{I}_2^*$-uniformly Cauchy sequences for double sequences of functions in $2$-normed spaces. Then, we show the relationships between these new concepts.

Keywords

Double sequence of functions $\mathcal{I}$-Convergence Uniformly convergence 2-normed spaces

References

  • [1] M. Arslan, E. D¨undar, I-Convergence and I-Cauchy Sequence of Functions In 2-Normed Spaces, Konuralp Journal of Mathematics, 6(1) (2018), 57–62.
  • [2] M. Arslan, E. D¨undar, On I-Convergence of sequences of functions in 2-normed spaces, Southeast Asian Bulletin of Mathematics, 42 (2018), 491–502.
  • [3] M. Arslan, E. D¨undar, Rough convergence in 2-normed spaces, Bulletin of Mathematical Analysis and Applications, 10(3) (2018), 1–9.
  • [4] M. Arslan, E. D¨undar, On Rough Convergence in 2-Normed Spaces and Some Properties Filomat 33(16) (2019), 5077– 5086.
  • [5] V. Bala´z, J. C˘ erven˘ansky´, P. Kostyrko, T. S˘ala´t, I-convergence and I-continuity of real functions, Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University, Nitra, 5 (2004), 43–50.
  • [6] M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328(1) (2007), 715-729.
  • [7] H. C¸ akallı and S. Ersan, New types of continuity in 2-normed spaces, Filomat, 30(3) (2016), 525–532.
  • [8] P. Das, P. Kostyrko, W. Wilczy´nski, P. Malik, I and I∗-convergence of double sequences, Math. Slovaca, 58 (2008), No. 5, 605–620.
  • [9] P. Das, P. Malik, On extremal I-limit points of double sequences, Tatra Mt. Math. Publ. 40 (2008), 91–102.
  • [10] E. D¨undar, B. Altay, I2-convergence of double sequences of functions, Electronic Journal of Mathematical Analysis and Applications, 3(1) (2015), 111–121.
  • [11] E. D¨undar, B. Altay, I2-uniform convergence of double sequences of functions, Filomat, 30(5) (2016), 1273–1281.
  • [12] E. D¨undar, On some results of I2-convergence of double sequences of functions, Mathematical Analysis Sciences and Applications E-notes, 3(1) (2015), 44–52.
  • [13] E. D¨undar, M. Arslan, S. Yeg¨ul, On I-Uniform convergence of sequences of functions in 2-normed spaces, Rocky Mountain Journal of Mathematics 50(5) (2020), 1637–1646.
  • [14] E. D¨undar, U. Ulusu, N. Pancaroˇglu, Strongly I2-lacunary convergence and I2-lacunary Cauchy double sequences of sets, The Aligarh Bulletin of Mathematisc, 35(1-2) (2016), 1-15.
  • [15] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
  • [16] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313.
  • [17] S. G¨ahler, 2-metrische R¨aume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148.
  • [18] S. G¨ahler, 2-normed spaces, Math. Nachr. 28 (1964), 1–43.
  • [19] F. Gezer, S. Karakus¸, I and I∗ convergent function sequences, Math. Commun. 10 (2005), 71-80.
  • [20] A. G¨okhan, M. G¨ung¨or and M. Et, Statistical convergence of double sequences of real-valued functions, Int. Math. Forum, 2(8) (2007), 365-374.
  • [21] H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (10) (2001), 631–639.
  • [22] H. Gunawan, M. Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math. 27 (3) (2001), 321–329.
  • [23] M. G¨urdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2 (1) (2004), 107–113.
  • [24] M. G¨urdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math. 33 (2009), 257–264.
  • [25] M. G¨urdal, I. Ac¸ık, On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11(2) (2008), 349–354.
  • [26] M. G¨urdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4(1) (2006), 85–91.
  • [27] P. Kostyrko, T. ˘ Sal´at, W. Wilczy´nski, I-convergence, Real Anal. Exchange 26 (2) (2000), 669–686.
  • [28] M. Mursaleen, S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62(1) (2012), 49–62.
  • [29] M. Mursaleen, A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca 61(6) (2011), 933–940.
  • [30] F. Nuray, W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513–527.
  • [31] F. Nuray, E. D¨undar, U. Ulusu, Wijsman I2-convergence of double sequences of cosed sets, Pure and Applied Mathematics Letters, 2 (2014), 35-39.
  • [32] S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces, Int. J. Math. Math. Sci. 2011 (2011), 10 pages. doi:10.1155/2011/517841.
  • [33] E. Savas¸, M. G¨urdal, Ideal Convergent Function Sequences in Random 2-Normed Spaces, Filomat, 30(3) (2016), 557–567.
  • [34] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
  • [35] A. Sharma, K. Kumar, Statistical convergence in probabilistic 2-normed spaces, Mathematical Sciences, 2(4) (2008), 373–390.
  • [36] A. S¸ ahiner, M. G¨urdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11(5) (2007), 1477–1484.
  • [37] B.C. Tripathy, M. Sen, S. Nath, I-convergence in probabilistic n-normed space, Soft Comput. 16(6) (2012), 1021–1027.
  • [38] U. Ulusu, F. Nuray, E. D¨undar, I-limit and I-cluster points for functions defined on amenable semigroups, Fundamental Journal of Mathematics and Applications, 4(2) (2021), 45-48.
  • [39] S. Yeg¨ul, E. D¨undar, On Statistical convergence of sequences of functions in 2-normed spaces, Journal of Classical Analysis, 10(1) (2017), 49–57.
  • [40] S. Yeg¨ul, E. D¨undar, Statistical convergence of double sequences of functions and some properties in 2-normed spaces, Facta Universitatis, Series Mathematics and Informatics, 33(5) (2018), 705–719.
  • [41] S. Yeg¨ul, E. D¨undar, I2-convergence of double sequences of functions in 2-normed spaces, Universal Journal of Mathematics and Applications 2(3) (2019) 130–137.
  • [42] S. Yeg¨ul, E. D¨undar, On I2-convergence and I2-Cauchy double sequences of functions in 2-normed spaces, Facta Universitatis Series Mathematics and Informatics 35(3) (2020) 801–814.

Year 2022, Volume: 5 Issue: 3, 150 - 160, 30.09.2022

Sevim Yegül Güzey , Erdinç Dündar , Mukaddes Arslan

https://doi.org/10.33434/cams.1177174

Abstract

References

  • [1] M. Arslan, E. D¨undar, I-Convergence and I-Cauchy Sequence of Functions In 2-Normed Spaces, Konuralp Journal of Mathematics, 6(1) (2018), 57–62.
  • [2] M. Arslan, E. D¨undar, On I-Convergence of sequences of functions in 2-normed spaces, Southeast Asian Bulletin of Mathematics, 42 (2018), 491–502.
  • [3] M. Arslan, E. D¨undar, Rough convergence in 2-normed spaces, Bulletin of Mathematical Analysis and Applications, 10(3) (2018), 1–9.
  • [4] M. Arslan, E. D¨undar, On Rough Convergence in 2-Normed Spaces and Some Properties Filomat 33(16) (2019), 5077– 5086.
  • [5] V. Bala´z, J. C˘ erven˘ansky´, P. Kostyrko, T. S˘ala´t, I-convergence and I-continuity of real functions, Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University, Nitra, 5 (2004), 43–50.
  • [6] M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328(1) (2007), 715-729.
  • [7] H. C¸ akallı and S. Ersan, New types of continuity in 2-normed spaces, Filomat, 30(3) (2016), 525–532.
  • [8] P. Das, P. Kostyrko, W. Wilczy´nski, P. Malik, I and I∗-convergence of double sequences, Math. Slovaca, 58 (2008), No. 5, 605–620.
  • [9] P. Das, P. Malik, On extremal I-limit points of double sequences, Tatra Mt. Math. Publ. 40 (2008), 91–102.
  • [10] E. D¨undar, B. Altay, I2-convergence of double sequences of functions, Electronic Journal of Mathematical Analysis and Applications, 3(1) (2015), 111–121.
  • [11] E. D¨undar, B. Altay, I2-uniform convergence of double sequences of functions, Filomat, 30(5) (2016), 1273–1281.
  • [12] E. D¨undar, On some results of I2-convergence of double sequences of functions, Mathematical Analysis Sciences and Applications E-notes, 3(1) (2015), 44–52.
  • [13] E. D¨undar, M. Arslan, S. Yeg¨ul, On I-Uniform convergence of sequences of functions in 2-normed spaces, Rocky Mountain Journal of Mathematics 50(5) (2020), 1637–1646.
  • [14] E. D¨undar, U. Ulusu, N. Pancaroˇglu, Strongly I2-lacunary convergence and I2-lacunary Cauchy double sequences of sets, The Aligarh Bulletin of Mathematisc, 35(1-2) (2016), 1-15.
  • [15] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
  • [16] J.A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313.
  • [17] S. G¨ahler, 2-metrische R¨aume und ihre topologische struktur, Math. Nachr. 26 (1963), 115–148.
  • [18] S. G¨ahler, 2-normed spaces, Math. Nachr. 28 (1964), 1–43.
  • [19] F. Gezer, S. Karakus¸, I and I∗ convergent function sequences, Math. Commun. 10 (2005), 71-80.
  • [20] A. G¨okhan, M. G¨ung¨or and M. Et, Statistical convergence of double sequences of real-valued functions, Int. Math. Forum, 2(8) (2007), 365-374.
  • [21] H. Gunawan, M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27 (10) (2001), 631–639.
  • [22] H. Gunawan, M. Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math. 27 (3) (2001), 321–329.
  • [23] M. G¨urdal, S. Pehlivan, The statistical convergence in 2-Banach spaces, Thai J. Math. 2 (1) (2004), 107–113.
  • [24] M. G¨urdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bull. Math. 33 (2009), 257–264.
  • [25] M. G¨urdal, I. Ac¸ık, On I-Cauchy sequences in 2-normed spaces, Math. Inequal. Appl. 11(2) (2008), 349–354.
  • [26] M. G¨urdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4(1) (2006), 85–91.
  • [27] P. Kostyrko, T. ˘ Sal´at, W. Wilczy´nski, I-convergence, Real Anal. Exchange 26 (2) (2000), 669–686.
  • [28] M. Mursaleen, S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Math. Slovaca 62(1) (2012), 49–62.
  • [29] M. Mursaleen, A. Alotaibi, On I-convergence in random 2-normed spaces, Math. Slovaca 61(6) (2011), 933–940.
  • [30] F. Nuray, W.H. Ruckle, Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245 (2000), 513–527.
  • [31] F. Nuray, E. D¨undar, U. Ulusu, Wijsman I2-convergence of double sequences of cosed sets, Pure and Applied Mathematics Letters, 2 (2014), 35-39.
  • [32] S. Sarabadan, S. Talebi, Statistical convergence and ideal convergence of sequences of functions in 2-normed spaces, Int. J. Math. Math. Sci. 2011 (2011), 10 pages. doi:10.1155/2011/517841.
  • [33] E. Savas¸, M. G¨urdal, Ideal Convergent Function Sequences in Random 2-Normed Spaces, Filomat, 30(3) (2016), 557–567.
  • [34] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
  • [35] A. Sharma, K. Kumar, Statistical convergence in probabilistic 2-normed spaces, Mathematical Sciences, 2(4) (2008), 373–390.
  • [36] A. S¸ ahiner, M. G¨urdal, S. Saltan, H. Gunawan, Ideal convergence in 2-normed spaces, Taiwanese J. Math. 11(5) (2007), 1477–1484.
  • [37] B.C. Tripathy, M. Sen, S. Nath, I-convergence in probabilistic n-normed space, Soft Comput. 16(6) (2012), 1021–1027.
  • [38] U. Ulusu, F. Nuray, E. D¨undar, I-limit and I-cluster points for functions defined on amenable semigroups, Fundamental Journal of Mathematics and Applications, 4(2) (2021), 45-48.
  • [39] S. Yeg¨ul, E. D¨undar, On Statistical convergence of sequences of functions in 2-normed spaces, Journal of Classical Analysis, 10(1) (2017), 49–57.
  • [40] S. Yeg¨ul, E. D¨undar, Statistical convergence of double sequences of functions and some properties in 2-normed spaces, Facta Universitatis, Series Mathematics and Informatics, 33(5) (2018), 705–719.
  • [41] S. Yeg¨ul, E. D¨undar, I2-convergence of double sequences of functions in 2-normed spaces, Universal Journal of Mathematics and Applications 2(3) (2019) 130–137.
  • [42] S. Yeg¨ul, E. D¨undar, On I2-convergence and I2-Cauchy double sequences of functions in 2-normed spaces, Facta Universitatis Series Mathematics and Informatics 35(3) (2020) 801–814.
There are 42 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sevim Yegül Güzey

Erdinç Dündar 0000-0002-0545-7486

Mukaddes Arslan 0000-0002-5798-670X

Publication Date September 30, 2022
Submission Date September 19, 2022
Acceptance Date September 30, 2022
Published in Issue Year 2022 Volume: 5 Issue: 3

Cite

APA Yegül Güzey, S., Dündar, E., & Arslan, M. (2022). $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces. Communications in Advanced Mathematical Sciences, 5(3), 150-160. https://doi.org/10.33434/cams.1177174
AMA Yegül Güzey S, Dündar E, Arslan M. $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces. Communications in Advanced Mathematical Sciences. September 2022;5(3):150-160. doi:10.33434/cams.1177174
Chicago Yegül Güzey, Sevim, Erdinç Dündar, and Mukaddes Arslan. “$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces”. Communications in Advanced Mathematical Sciences 5, no. 3 (September 2022): 150-60. https://doi.org/10.33434/cams.1177174.
EndNote Yegül Güzey S, Dündar E, Arslan M (September 1, 2022) $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces. Communications in Advanced Mathematical Sciences 5 3 150–160.
IEEE S. Yegül Güzey, E. Dündar, and M. Arslan, “$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces”, Communications in Advanced Mathematical Sciences, vol. 5, no. 3, pp. 150–160, 2022, doi: 10.33434/cams.1177174.
ISNAD Yegül Güzey, Sevim et al. “$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces”. Communications in Advanced Mathematical Sciences 5/3 (September2022), 150-160. https://doi.org/10.33434/cams.1177174.
JAMA Yegül Güzey S, Dündar E, Arslan M. $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces. Communications in Advanced Mathematical Sciences. 2022;5:150–160.
MLA Yegül Güzey, Sevim et al. “$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces”. Communications in Advanced Mathematical Sciences, vol. 5, no. 3, 2022, pp. 150-6, doi:10.33434/cams.1177174.
Vancouver Yegül Güzey S, Dündar E, Arslan M. $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces. Communications in Advanced Mathematical Sciences. 2022;5(3):150-6.
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