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Year 2024, , 1 - 11, 28.01.2024

Mukaddes Arslan , Ramazan Sunar

https://doi.org/10.36753/mathenot.1308678

Abstract

References

  • [1] Gähler, S.: 2-metrische Räume und ihre topologische struktur. Mathematische Nachrichten. 26, 115-148 (1963).
  • [2] Gähler, S.: Lineare 2-normierte Räume. Mathematische Nachrichten. 28, 1-43 (1964).
  • [3] Gürdal, M., Pehlivan S.: Statistical convergence in 2-normed spaces. Southeast Asian Bulletin of Mathematics. 33, 257-264 (2009).
  • [4] Şahiner, A., Gürdal, M., Saltan, S., Gunawan, H.: Ideal convergence in 2-normed spaces. Taiwanese Journal of Mathematics. 11(5), 1477-1484 (2007). https://doi.org/10.11650/twjm/1500404879
  • [5] Gürdal, M.: On ideal convergent sequences in 2-normed spaces. Thai Journal of Mathematics. 4(1), 85-91 (2006).
  • [6] Gürdal, M., Açık, I.: On I-Cauchy sequences in 2-normed spaces. Mathematical Inequalities and Applications. 11(2), 349-354 (2008).
  • [7] Çakallı, H., Ersan, S.: New types of continuity in 2-normed spaces. Filomat. 30(3), 525-532 (2016). https://doi.org/10.2298/FIL1603525C
  • [8] Misiak, A.: n-inner product spaces. Mathematische Nachrichten. 140(1), 299-319 (1989).
  • [9] Gunawan, H.: The space of p-summable sequences and its natural n-norms. Bulletin of the Australian Mathematical Society. 64, 137-147 (2001).
  • [10] Gunawan, H.: Inner products on n-inner product spaces. Soochow Journal of Mathematics. 28(4), 389-398 (2002).
  • [11] Reddy, B.S.: Statistical convergence in n-normed spaces. International Mathematical Forum. 24, 1185-1193 (2010).
  • [12] Hazarika, B., Sava¸s, E.: $\lambda$-statistical convergence in n-normed spaces. An. St. Univ. Ovidius Constanta, Ser. Mat. 21(2), 141-153 (2013). https://doi.org/10.2478/auom-2013-0028
  • [13] Gürdal, M., ¸Sahiner, A.: Ideal convergence in n-normal spaces and some new sequence spaces via n-norm. Journal of Fundamental Sciences. 4(1), 233-244 (2008). https://doi.org/10.11113/mjfas.v4n1.32
  • [14] Phu, H. X.: Rough convergence in normed linear spaces. Numerical Functional Analysis and Optimization. 22, 199-222 (2001). https://doi.org/10.1081/NFA-100103794
  • [15] Phu, H. X.: Rough continuity of linear operators. Numerical Functional Analysis and Optimization. 23, 139-146 (2002). https://doi.org/10.1081/NFA-120003675
  • [16] Phu, H. X.: Rough convergence in infinite dimensional normed spaces. Numerical Functional Analysis and Optimization. 24, 285-301 (2003). https://doi.org/10.1081/NFA-120022923
  • [17] Aytar, S.: Rough statistical convergence, Numerical Functional Analysis and Optimization. 29(3-4), 291-303 (2008). https://doi.org/10.1080/01630560802001064
  • [18] Aytar, S.: The rough limit set and the core of a real sequence. Numerical Functional Analysis and Optimization. 29(3-4), 283-290 (2008). https://doi.org/10.1080/01630560802001056
  • [19] Arslan, M., Dündar, E.: Rough convergence in 2-normed spaces. Bulletin of Mathematical Analysis and Applications. 10(3), 1-9 (2018).
  • [20] Arslan, M., Dündar, E.: On rough convergence in 2-normed spaces and some properties. Filomat. 33(16), 5077-5086 (2019). https://doi.org/10.2298/FIL1916077A
  • [21] Arslan, M., Dündar, E.: Rough statistical convergence in 2-normed spaces. Honam Mathematical Journal. 43(3), 417-431 (2021). https://doi.org/10.5831/HMJ.2021.43.3.417
  • [22] Arslan, M., Dündar, E.: Rough statistical cluster points in 2-normed spaces. Thai Journal of Mathematics. 20(3), 1419-1429 (2022).
  • [23] Sunar, R., Arslan, M.: Rough convergence in n-normed spaces. (Submitted).
  • [24] Pringsheim, A.: Elementare Theorie der unendliche Doppelreihen. Sitsungs berichte der Math. Akad. derWissenscha Mnch. Ber. 7, 101-153 (1898).
  • [25] Pringsheim, A.: Zur theorie der zweifach unendlichen zahlenfolgen.Mathematische Annalen. 53, 289321 (1900).
  • [26] Hardy, G. H.: On the convergence of certain multiple series. Proc. Cambridge Philos. Soc. 19, 86-95 (1916-1919). https://doi.org/10.1112/plms/s2-1.1.124
  • [27] Das, P., Malik, P.: On the statistical and I-variation of double sequences. Real Analysis Exchange. 33, 351-364 (2007).
  • [28] Das, P., Malik, P.: On extremal I-limit points of double sequences. Tatra Mountains Mathematical Publications. 40, 91-102 (2008).
  • [29] Das, P., Kostyrko, P., WilczynskiW., Malik P.: I and I-convergence of double sequences. Mathematica Slovaca. 58, 605-620 (2008). https://doi.org/10.2478/s12175-008-0096-x
  • [30] Khan, V. A., Alshlool, K.M., Abdullah, S.A., Rababah, R.K., Ahmad, A.: Some new classes of paranorm ideal convergent double sequences of sigma-bounded variation over n-normed spaces. Cogent Mathematics & Statistics. 5(1), 1460029 (2018). https://doi.org/10.1080/25742558.2018.1460029
  • [31] Mursaleen, M., Edely, O. H. H.: Statistical convergence of double sequences. Journal of Mathematical Analysis and Applications. 288, 223-231 (2003). https://doi.org/10.1016/j.jmaa.2003.08.004
  • [32] Patterson, R. F.: Double sequence core theorems.International Journal of Mathematics and Mathematical Sciences. 22(4), 785-793 (1999).
  • [33] Malik, P., Maity, M.: On rough convergence of double sequences in normed linear spaces. Bulletin of the Allahabad Mathematical Society. 28(1), 89-99 (2013).
  • [34] Dündar, E., Çakan, C.: Rough I-convergence. Demonstratio Mathematica. 2(1), 45-51 (2014). https://doi.org/10.2478/dema-2014-0051
  • [35] Dündar, E., Çakan, C.: Rough convergence of double sequences. Gulf Journal of Mathematics. 47(3), 638-651 (2014).
  • [36] Kişi, Ö., Dündar, E.: Rough I2-lacunary statistical convergence of double sequences. Journal of Inequalities and Applications. 2018(230), 1-16 (2018). https://doi.org/10.1186/s13660-018-1831-7
  • [37] Gunawan, H., Mashadi, M.: On n-normed spaces. International Journal of Mathematics and Mathematical Sciences. 27 (10), 631-639 (2001). https://doi.org/10.1155/S0161171201010675

Rough Convergence of Double Sequences in $n-$Normed Spaces

Year 2024, , 1 - 11, 28.01.2024

Mukaddes Arslan , Ramazan Sunar

https://doi.org/10.36753/mathenot.1308678

Abstract

In this study, we introduced the concepts of rough convergence, rough Cauchy double sequence, and the set of rough limit points of a double sequence, as well as the rough convergence criteria associated with this set in $n$-normed spaces. Later, we proved that this set is both closed and convex. Finally, we presented the relationships between rough convergence and rough Cauchy double sequence in $n$-normed spaces.

Keywords

Cauchy sequence Double sequence $n-$normed space Rough convergence

References

  • [1] Gähler, S.: 2-metrische Räume und ihre topologische struktur. Mathematische Nachrichten. 26, 115-148 (1963).
  • [2] Gähler, S.: Lineare 2-normierte Räume. Mathematische Nachrichten. 28, 1-43 (1964).
  • [3] Gürdal, M., Pehlivan S.: Statistical convergence in 2-normed spaces. Southeast Asian Bulletin of Mathematics. 33, 257-264 (2009).
  • [4] Şahiner, A., Gürdal, M., Saltan, S., Gunawan, H.: Ideal convergence in 2-normed spaces. Taiwanese Journal of Mathematics. 11(5), 1477-1484 (2007). https://doi.org/10.11650/twjm/1500404879
  • [5] Gürdal, M.: On ideal convergent sequences in 2-normed spaces. Thai Journal of Mathematics. 4(1), 85-91 (2006).
  • [6] Gürdal, M., Açık, I.: On I-Cauchy sequences in 2-normed spaces. Mathematical Inequalities and Applications. 11(2), 349-354 (2008).
  • [7] Çakallı, H., Ersan, S.: New types of continuity in 2-normed spaces. Filomat. 30(3), 525-532 (2016). https://doi.org/10.2298/FIL1603525C
  • [8] Misiak, A.: n-inner product spaces. Mathematische Nachrichten. 140(1), 299-319 (1989).
  • [9] Gunawan, H.: The space of p-summable sequences and its natural n-norms. Bulletin of the Australian Mathematical Society. 64, 137-147 (2001).
  • [10] Gunawan, H.: Inner products on n-inner product spaces. Soochow Journal of Mathematics. 28(4), 389-398 (2002).
  • [11] Reddy, B.S.: Statistical convergence in n-normed spaces. International Mathematical Forum. 24, 1185-1193 (2010).
  • [12] Hazarika, B., Sava¸s, E.: $\lambda$-statistical convergence in n-normed spaces. An. St. Univ. Ovidius Constanta, Ser. Mat. 21(2), 141-153 (2013). https://doi.org/10.2478/auom-2013-0028
  • [13] Gürdal, M., ¸Sahiner, A.: Ideal convergence in n-normal spaces and some new sequence spaces via n-norm. Journal of Fundamental Sciences. 4(1), 233-244 (2008). https://doi.org/10.11113/mjfas.v4n1.32
  • [14] Phu, H. X.: Rough convergence in normed linear spaces. Numerical Functional Analysis and Optimization. 22, 199-222 (2001). https://doi.org/10.1081/NFA-100103794
  • [15] Phu, H. X.: Rough continuity of linear operators. Numerical Functional Analysis and Optimization. 23, 139-146 (2002). https://doi.org/10.1081/NFA-120003675
  • [16] Phu, H. X.: Rough convergence in infinite dimensional normed spaces. Numerical Functional Analysis and Optimization. 24, 285-301 (2003). https://doi.org/10.1081/NFA-120022923
  • [17] Aytar, S.: Rough statistical convergence, Numerical Functional Analysis and Optimization. 29(3-4), 291-303 (2008). https://doi.org/10.1080/01630560802001064
  • [18] Aytar, S.: The rough limit set and the core of a real sequence. Numerical Functional Analysis and Optimization. 29(3-4), 283-290 (2008). https://doi.org/10.1080/01630560802001056
  • [19] Arslan, M., Dündar, E.: Rough convergence in 2-normed spaces. Bulletin of Mathematical Analysis and Applications. 10(3), 1-9 (2018).
  • [20] Arslan, M., Dündar, E.: On rough convergence in 2-normed spaces and some properties. Filomat. 33(16), 5077-5086 (2019). https://doi.org/10.2298/FIL1916077A
  • [21] Arslan, M., Dündar, E.: Rough statistical convergence in 2-normed spaces. Honam Mathematical Journal. 43(3), 417-431 (2021). https://doi.org/10.5831/HMJ.2021.43.3.417
  • [22] Arslan, M., Dündar, E.: Rough statistical cluster points in 2-normed spaces. Thai Journal of Mathematics. 20(3), 1419-1429 (2022).
  • [23] Sunar, R., Arslan, M.: Rough convergence in n-normed spaces. (Submitted).
  • [24] Pringsheim, A.: Elementare Theorie der unendliche Doppelreihen. Sitsungs berichte der Math. Akad. derWissenscha Mnch. Ber. 7, 101-153 (1898).
  • [25] Pringsheim, A.: Zur theorie der zweifach unendlichen zahlenfolgen.Mathematische Annalen. 53, 289321 (1900).
  • [26] Hardy, G. H.: On the convergence of certain multiple series. Proc. Cambridge Philos. Soc. 19, 86-95 (1916-1919). https://doi.org/10.1112/plms/s2-1.1.124
  • [27] Das, P., Malik, P.: On the statistical and I-variation of double sequences. Real Analysis Exchange. 33, 351-364 (2007).
  • [28] Das, P., Malik, P.: On extremal I-limit points of double sequences. Tatra Mountains Mathematical Publications. 40, 91-102 (2008).
  • [29] Das, P., Kostyrko, P., WilczynskiW., Malik P.: I and I-convergence of double sequences. Mathematica Slovaca. 58, 605-620 (2008). https://doi.org/10.2478/s12175-008-0096-x
  • [30] Khan, V. A., Alshlool, K.M., Abdullah, S.A., Rababah, R.K., Ahmad, A.: Some new classes of paranorm ideal convergent double sequences of sigma-bounded variation over n-normed spaces. Cogent Mathematics & Statistics. 5(1), 1460029 (2018). https://doi.org/10.1080/25742558.2018.1460029
  • [31] Mursaleen, M., Edely, O. H. H.: Statistical convergence of double sequences. Journal of Mathematical Analysis and Applications. 288, 223-231 (2003). https://doi.org/10.1016/j.jmaa.2003.08.004
  • [32] Patterson, R. F.: Double sequence core theorems.International Journal of Mathematics and Mathematical Sciences. 22(4), 785-793 (1999).
  • [33] Malik, P., Maity, M.: On rough convergence of double sequences in normed linear spaces. Bulletin of the Allahabad Mathematical Society. 28(1), 89-99 (2013).
  • [34] Dündar, E., Çakan, C.: Rough I-convergence. Demonstratio Mathematica. 2(1), 45-51 (2014). https://doi.org/10.2478/dema-2014-0051
  • [35] Dündar, E., Çakan, C.: Rough convergence of double sequences. Gulf Journal of Mathematics. 47(3), 638-651 (2014).
  • [36] Kişi, Ö., Dündar, E.: Rough I2-lacunary statistical convergence of double sequences. Journal of Inequalities and Applications. 2018(230), 1-16 (2018). https://doi.org/10.1186/s13660-018-1831-7
  • [37] Gunawan, H., Mashadi, M.: On n-normed spaces. International Journal of Mathematics and Mathematical Sciences. 27 (10), 631-639 (2001). https://doi.org/10.1155/S0161171201010675
There are 37 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mukaddes Arslan 0000-0002-5798-670X

Ramazan Sunar 0000-0001-8107-5618

Early Pub Date November 2, 2023
Publication Date January 28, 2024
Submission Date June 1, 2023
Acceptance Date July 13, 2023
Published in Issue Year 2024

Cite

APA Arslan, M., & Sunar, R. (2024). Rough Convergence of Double Sequences in $n-$Normed Spaces. Mathematical Sciences and Applications E-Notes, 12(1), 1-11. https://doi.org/10.36753/mathenot.1308678
AMA Arslan M, Sunar R. Rough Convergence of Double Sequences in $n-$Normed Spaces. Math. Sci. Appl. E-Notes. January 2024;12(1):1-11. doi:10.36753/mathenot.1308678
Chicago Arslan, Mukaddes, and Ramazan Sunar. “Rough Convergence of Double Sequences in $n-$Normed Spaces”. Mathematical Sciences and Applications E-Notes 12, no. 1 (January 2024): 1-11. https://doi.org/10.36753/mathenot.1308678.
EndNote Arslan M, Sunar R (January 1, 2024) Rough Convergence of Double Sequences in $n-$Normed Spaces. Mathematical Sciences and Applications E-Notes 12 1 1–11.
IEEE M. Arslan and R. Sunar, “Rough Convergence of Double Sequences in $n-$Normed Spaces”, Math. Sci. Appl. E-Notes, vol. 12, no. 1, pp. 1–11, 2024, doi: 10.36753/mathenot.1308678.
ISNAD Arslan, Mukaddes - Sunar, Ramazan. “Rough Convergence of Double Sequences in $n-$Normed Spaces”. Mathematical Sciences and Applications E-Notes 12/1 (January 2024), 1-11. https://doi.org/10.36753/mathenot.1308678.
JAMA Arslan M, Sunar R. Rough Convergence of Double Sequences in $n-$Normed Spaces. Math. Sci. Appl. E-Notes. 2024;12:1–11.
MLA Arslan, Mukaddes and Ramazan Sunar. “Rough Convergence of Double Sequences in $n-$Normed Spaces”. Mathematical Sciences and Applications E-Notes, vol. 12, no. 1, 2024, pp. 1-11, doi:10.36753/mathenot.1308678.
Vancouver Arslan M, Sunar R. Rough Convergence of Double Sequences in $n-$Normed Spaces. Math. Sci. Appl. E-Notes. 2024;12(1):1-11.

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