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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
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.

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
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.

Details

Primary Language English
Subjects Mathematics
Journal Section Articles
Authors

Sevim YEGÜL GÜZEY>
It is not affiliated with an institution
0000-0001-8301-0252
Türkiye


Erdinç DÜNDAR> (Primary Author)
AFYON KOCATEPE ÜNİVERSİTESİ
0000-0002-0545-7486
Türkiye


Mukaddes ARSLAN>
It is not affiliated with an institution
0000-0002-5798-670X
Türkiye

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

Bibtex @research article { cams1177174, journal = {Communications in Advanced Mathematical Sciences}, issn = {2651-4001}, address = {}, publisher = {Emrah Evren KARA}, year = {2022}, volume = {5}, number = {3}, pages = {150 - 160}, doi = {10.33434/cams.1177174}, title = {\$\\mathcal\{I\}\_2\$-Uniform Convergence of Double Sequences of Functions In \$2\$-Normed Spaces}, key = {cite}, author = {Yegül Güzey, Sevim and Dündar, Erdinç and Arslan, Mukaddes} }
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 . DOI: 10.33434/cams.1177174
MLA 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 5 (2022 ): 150-160 <https://dergipark.org.tr/en/pub/cams/issue/72815/1177174>
Chicago 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 5 (2022 ): 150-160
RIS TY - JOUR T1 - $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces AU - SevimYegül Güzey, ErdinçDündar, MukaddesArslan Y1 - 2022 PY - 2022 N1 - doi: 10.33434/cams.1177174 DO - 10.33434/cams.1177174 T2 - Communications in Advanced Mathematical Sciences JF - Journal JO - JOR SP - 150 EP - 160 VL - 5 IS - 3 SN - 2651-4001- M3 - doi: 10.33434/cams.1177174 UR - https://doi.org/10.33434/cams.1177174 Y2 - 2022 ER -
EndNote %0 Communications in Advanced Mathematical Sciences $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces %A Sevim Yegül Güzey , Erdinç Dündar , Mukaddes Arslan %T $\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces %D 2022 %J Communications in Advanced Mathematical Sciences %P 2651-4001- %V 5 %N 3 %R doi: 10.33434/cams.1177174 %U 10.33434/cams.1177174
ISNAD Yegül Güzey, Sevim , Dündar, Erdinç , Arslan, Mukaddes . "$\mathcal{I}_2$-Uniform Convergence of Double Sequences of Functions In $2$-Normed Spaces". Communications in Advanced Mathematical Sciences 5 / 3 (September 2022): 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. 2022; 5(3): 150-160.
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): 150-160.
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, Sep. 2022, doi:10.33434/cams.1177174
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