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Year 2024, Volume: 12 Issue: 1, 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, Volume: 12 Issue: 1, 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 Volume: 12 Issue: 1

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 (January2024), 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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