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Statistical Convergence of Nets Through Directed Sets

Year 2019, , 79 - 84, 28.06.2019
https://doi.org/10.32323/ujma.539127

Abstract

The concept of statistical convergence based on asymptotic density is introduced in this article through nets. Some possible extensions of classical results for statistical convergence of sequences are obtained in this article, with extensions to nets.

References

  • [1] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topol. Appl., 159 (2012), 1887-1893.
  • [2] A. Alotaibi, A. M. Alroqi, Statistical convergence in a paranorned space, J. Inequal. Appl., 39 (2012), 6 pages, doi: 10.1186/1029-242X-2012-39 .
  • [3] B. Bilalov, T. Nazarova, On statistical convergence in metric spaces, J. Math. Res., 7(1) (2015), 37-43.
  • [4] B. Bilalov, T. Nazarova, On statistical type convergence in uniform spaces, Bull. of the Iranian Math. Soc., 42(4) (2016), 975-986.
  • [5] R. C. Buck, Generalized asymptotic density, Amer. J. Math., 75 (1953), 335-346.
  • [6] H. Cakalli, On statistical convergence in topological groups, Pure Appl. Math. Sci., 43 (1996), 27-31.
  • [7] E. D¨undar, Y. Sever, Multipliers for bounded statistical convergence of double sequences, Int. Math. Forum., 7(52) (2012), 2581-2587.
  • [8] E. D¨undar, U. Ulusu, B. Aydin, I2-lacunary statistical convergence of double sequences of sets, Konuralp J. Math., 5(1) (2017), 1-10.
  • [9] E. D¨undar, U. Ulusu, F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inform., 27(2) (2018), 161-169.
  • [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
  • [11] J. A. Fridy, On statistical convergence, Anal., 5 (1985), 301-313.
  • [12] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187-1192.
  • [13] J. L. Kelly, General topology, Springer, (1975).
  • [14] E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu. Math., 928 (1991), 41-52.
  • [15] P. Kostyrko, W. Wilcznski, T. Salat, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
  • [16] B. K. Lahiri, P. Das, I and I-convergence in topological spaces, Math. Bohem., 130(2) (2005), 153-160.
  • [17] B. K. Lahiri, P. Das, I and I-convergence of nets, Real Anal. Exchange, 33(2) (2007-2008), 431-442.
  • [18] S. Loganathan, C. G. Moorthy, A net convergence for Schauder double bases, Asian-Eur. J. Math., 9(1) (2016), 1650010.
  • [19] S. Loganathan, C. G. Moorthy, Block convergence of series in topological vector spaces, J. Ana. Num. Theor., 4(1) (2016), 61-69.
  • [20] I. J. Maddox, Statistical convergence in a locally convex space, Math. Cambridge Phil. Soc., 104(1) (1988), 141-145.
  • [21] G. D. Maio, L. D. R. Kocinac, Statistical convergence in topology, Topol. Appl., 156 (2008), 28-45.
  • [22] C. G. Moorthy, A problem of Good on Hausdorff dimension, Mathematika, 39(2) (1992), 244-246.
  • [23] C. G. Moorthy, R. Vijaya, P. Venkatachalapathy, Hausdorff dimension of Cantor-like sets, Kyungpook Math. J., 32(2) (1992), 197-202.
  • [24] C. G. Moorthy, I. Raj, Weak convergence of fixed point iterations in metric spaces, J. Optimiz. Theory App., 4(2) (2013), 189-192.
  • [25] C. G. Moorthy, T. Ramasamy, Pringsheim convergence of double sequences for uniform boundedness principle, Asian-Eur. J. Math., 10(4) (2017), 1750080.
  • [26] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
  • [27] M. Mursaleen, O. H. H. Edely, Generalized statistical convergence, Inform. Sci., 162(3-4) (2004), 287-294.
  • [28] F. Nuray, U. Ulusu, E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20(7) (2016), 2883-2888.
  • [29] N. Pancarolu, E. Dündar, U. Ulusu, Asymptotically Isq -statistical equivalence of sequences of sets defined by a modulus functions, Sakarya Univ. J. Sci., 22(6) (2018), 1857-1862.
  • [30] D. Rath, B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure appl. Math., 25(4) (1994), 381-386.
  • [31] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
  • [32] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826-830.
  • [33] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1959), 361-375.
  • [34] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
  • [35] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequences of sets, Prog. Appl. Math., 4(2) (2012), 99-109.
  • [36] U. Ulusu, E. Dündar, I-Lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567-1574.
  • [37] U. Ulusu, Asymptotically ideal invariant equivalence, Creat. Math. Inform., 27(2) (2018), 215-220.
  • [38] S. Yegul, E. Dündar, On statistical convergence of sequences of functions in 2-normed spaces, J. Class. Anal., 10(1) (2017), 49-57.
  • [39] S. Yegul, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Universitatis, Ser. Math. and Infor., 33(5) (2018), 705-719.
Year 2019, , 79 - 84, 28.06.2019
https://doi.org/10.32323/ujma.539127

Abstract

References

  • [1] H. Albayrak, S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces, Topol. Appl., 159 (2012), 1887-1893.
  • [2] A. Alotaibi, A. M. Alroqi, Statistical convergence in a paranorned space, J. Inequal. Appl., 39 (2012), 6 pages, doi: 10.1186/1029-242X-2012-39 .
  • [3] B. Bilalov, T. Nazarova, On statistical convergence in metric spaces, J. Math. Res., 7(1) (2015), 37-43.
  • [4] B. Bilalov, T. Nazarova, On statistical type convergence in uniform spaces, Bull. of the Iranian Math. Soc., 42(4) (2016), 975-986.
  • [5] R. C. Buck, Generalized asymptotic density, Amer. J. Math., 75 (1953), 335-346.
  • [6] H. Cakalli, On statistical convergence in topological groups, Pure Appl. Math. Sci., 43 (1996), 27-31.
  • [7] E. D¨undar, Y. Sever, Multipliers for bounded statistical convergence of double sequences, Int. Math. Forum., 7(52) (2012), 2581-2587.
  • [8] E. D¨undar, U. Ulusu, B. Aydin, I2-lacunary statistical convergence of double sequences of sets, Konuralp J. Math., 5(1) (2017), 1-10.
  • [9] E. D¨undar, U. Ulusu, F. Nuray, On ideal invariant convergence of double sequences and some properties, Creat. Math. Inform., 27(2) (2018), 161-169.
  • [10] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244.
  • [11] J. A. Fridy, On statistical convergence, Anal., 5 (1985), 301-313.
  • [12] J. A. Fridy, Statistical limit points, Proc. Amer. Math. Soc., 118(4) (1993), 1187-1192.
  • [13] J. L. Kelly, General topology, Springer, (1975).
  • [14] E. Kolk, The statistical convergence in Banach spaces, Acta Comment. Univ. Tartu. Math., 928 (1991), 41-52.
  • [15] P. Kostyrko, W. Wilcznski, T. Salat, I-convergence, Real Anal. Exchange, 26(2) (2000), 669-686.
  • [16] B. K. Lahiri, P. Das, I and I-convergence in topological spaces, Math. Bohem., 130(2) (2005), 153-160.
  • [17] B. K. Lahiri, P. Das, I and I-convergence of nets, Real Anal. Exchange, 33(2) (2007-2008), 431-442.
  • [18] S. Loganathan, C. G. Moorthy, A net convergence for Schauder double bases, Asian-Eur. J. Math., 9(1) (2016), 1650010.
  • [19] S. Loganathan, C. G. Moorthy, Block convergence of series in topological vector spaces, J. Ana. Num. Theor., 4(1) (2016), 61-69.
  • [20] I. J. Maddox, Statistical convergence in a locally convex space, Math. Cambridge Phil. Soc., 104(1) (1988), 141-145.
  • [21] G. D. Maio, L. D. R. Kocinac, Statistical convergence in topology, Topol. Appl., 156 (2008), 28-45.
  • [22] C. G. Moorthy, A problem of Good on Hausdorff dimension, Mathematika, 39(2) (1992), 244-246.
  • [23] C. G. Moorthy, R. Vijaya, P. Venkatachalapathy, Hausdorff dimension of Cantor-like sets, Kyungpook Math. J., 32(2) (1992), 197-202.
  • [24] C. G. Moorthy, I. Raj, Weak convergence of fixed point iterations in metric spaces, J. Optimiz. Theory App., 4(2) (2013), 189-192.
  • [25] C. G. Moorthy, T. Ramasamy, Pringsheim convergence of double sequences for uniform boundedness principle, Asian-Eur. J. Math., 10(4) (2017), 1750080.
  • [26] M. Mursaleen, O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288 (2003), 223-231.
  • [27] M. Mursaleen, O. H. H. Edely, Generalized statistical convergence, Inform. Sci., 162(3-4) (2004), 287-294.
  • [28] F. Nuray, U. Ulusu, E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20(7) (2016), 2883-2888.
  • [29] N. Pancarolu, E. Dündar, U. Ulusu, Asymptotically Isq -statistical equivalence of sequences of sets defined by a modulus functions, Sakarya Univ. J. Sci., 22(6) (2018), 1857-1862.
  • [30] D. Rath, B. C. Tripathy, On statistically convergent and statistically Cauchy sequences, Indian J. Pure appl. Math., 25(4) (1994), 381-386.
  • [31] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
  • [32] E. Savas, P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett., 24(6) (2011), 826-830.
  • [33] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66(5) (1959), 361-375.
  • [34] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73-74.
  • [35] U. Ulusu, F. Nuray, Lacunary statistical convergence of sequences of sets, Prog. Appl. Math., 4(2) (2012), 99-109.
  • [36] U. Ulusu, E. Dündar, I-Lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567-1574.
  • [37] U. Ulusu, Asymptotically ideal invariant equivalence, Creat. Math. Inform., 27(2) (2018), 215-220.
  • [38] S. Yegul, E. Dündar, On statistical convergence of sequences of functions in 2-normed spaces, J. Class. Anal., 10(1) (2017), 49-57.
  • [39] S. Yegul, E. Dündar, Statistical Convergence of Double Sequences of Functions and Some Properties In 2-Normed Spaces, Facta Universitatis, Ser. Math. and Infor., 33(5) (2018), 705-719.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ar Murugan This is me 0000-0003-3119-7531

J. Dianavinnarasi This is me 0000-0003-3119-7531

C. Ganesa Moorthy 0000-0003-3119-7531

Publication Date June 28, 2019
Submission Date March 13, 2019
Acceptance Date May 3, 2019
Published in Issue Year 2019

Cite

APA Murugan, A., Dianavinnarasi, J., & Ganesa Moorthy, C. (2019). Statistical Convergence of Nets Through Directed Sets. Universal Journal of Mathematics and Applications, 2(2), 79-84. https://doi.org/10.32323/ujma.539127
AMA Murugan A, Dianavinnarasi J, Ganesa Moorthy C. Statistical Convergence of Nets Through Directed Sets. Univ. J. Math. Appl. June 2019;2(2):79-84. doi:10.32323/ujma.539127
Chicago Murugan, Ar, J. Dianavinnarasi, and C. Ganesa Moorthy. “Statistical Convergence of Nets Through Directed Sets”. Universal Journal of Mathematics and Applications 2, no. 2 (June 2019): 79-84. https://doi.org/10.32323/ujma.539127.
EndNote Murugan A, Dianavinnarasi J, Ganesa Moorthy C (June 1, 2019) Statistical Convergence of Nets Through Directed Sets. Universal Journal of Mathematics and Applications 2 2 79–84.
IEEE A. Murugan, J. Dianavinnarasi, and C. Ganesa Moorthy, “Statistical Convergence of Nets Through Directed Sets”, Univ. J. Math. Appl., vol. 2, no. 2, pp. 79–84, 2019, doi: 10.32323/ujma.539127.
ISNAD Murugan, Ar et al. “Statistical Convergence of Nets Through Directed Sets”. Universal Journal of Mathematics and Applications 2/2 (June 2019), 79-84. https://doi.org/10.32323/ujma.539127.
JAMA Murugan A, Dianavinnarasi J, Ganesa Moorthy C. Statistical Convergence of Nets Through Directed Sets. Univ. J. Math. Appl. 2019;2:79–84.
MLA Murugan, Ar et al. “Statistical Convergence of Nets Through Directed Sets”. Universal Journal of Mathematics and Applications, vol. 2, no. 2, 2019, pp. 79-84, doi:10.32323/ujma.539127.
Vancouver Murugan A, Dianavinnarasi J, Ganesa Moorthy C. Statistical Convergence of Nets Through Directed Sets. Univ. J. Math. Appl. 2019;2(2):79-84.

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