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I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space

Year 2019, Volume: 7 Issue: 1, 55 - 61, 15.04.2019

Şükran Konca , Ergin Genç Mehmet Küçükaslan

Abstract

In this work, we introduce the concepts of $\mathcal{I}$-statistical convergence and $\mathcal{I}$-lacunary statistical convergence of double sequences defined by weight functions in a locally solid Riesz space based on the notion of the ideal of subsets of $\mathbb{N}\times\mathbb{N}$. We also examine some inclusion relations of these concepts.


Keywords

Locally solid Riesz space $\mathcal{I}$-statistical convergence ideal convergence double sequences

References

  • [1] H. Albayrak, Statistical continuity and some convergence types in locally solid Riesz space. Phd Thesis, Institute of Science of S¨uleyman Demirel University, Isparta (2014).
  • [2] H. Albayrak and S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces. Top. Appl., 159(7) (2012), 1887-1893.
  • [3] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Amer. Math. Soc. No. 105 (2003).
  • [4] A. Alotaibi, B. Hazarika and S. A. Mohiuddine, On the ideal convergence of double sequences in locally solid Riesz spaces, Abst. Appl. Anal., 2014 Hindawi, Article ID 396254, (2014), 6 pages.
  • [5] T. M. Apostol, Some properties of completely multiplicative arithmetical functions, Amer. Math. Monthly., 78(3) (1971), 266-271.
  • [6] M. Balcerzak, P. Das, M. Filipczak and J. Swaczyna, Generalized kinds of density and associated ideals. Acta Math. Hungar. 147(1) (2015), 97-115.
  • [7] P. Das, P. Kostyrko, W. Wilczyski and P. Malik, I and I-convergence of double sequences. Math. Slovaca, 58(5) (2008), 605-620.
  • [8] K. Dems, On I-Cauchy sequence, Real Anal. Exchange, 30 (2004/2005), 123-128.
  • [9] E. Dundar, U. Ulusu and B. Aydın, I2-lacunary statistical convergence of double sequences of sets. Konuralp J. Math. 5(1) (2016), 1-10.
  • [10] E. Dundar, B. Altay, On some properties of I2-convergence and I2-Cauchy of double sequences. Gen. Math. Notes, 7(1) (2011), 1-12.
  • [11] E. Dundar and B. Altay, I2-convergence and I2-Cauchy of double sequences. Acta Math. Sci., 34(2) (2014), 343-353.
  • [12] B. Hazarika and A. Eşi, On ideal convergence in locally solid Riesz spaces using lacunary mean. Proc. Jangjeon Math. Soc., 19(2) (2016), 253-262.
  • [13] B. Hazarika, A. Eşi, Quasi-Slowly oscillating sequences in locally normal Riesz spaces. Int. J. Anal. Appl., 15(2) (2017), 229-237. DOI: 10.28924/2291- 8639-15-2017-229.
  • [14] H. Fast, Sur la convergence statistique. Colloq. Math., 2(3-4) (1951), 241-244.
  • [15] H. Freudenthal, Teilweise geordnete Moduln, K. Akademie van Wetenschappen, Afdeeling Natuurkunde. Proc. Sec. Sci., 39 (1936), 647-657.
  • [16] B. Hazarika, S. A. Mohiuddine and M. Mursaleen, Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces. Iranian J. Sci. Tech., 38(1) (2014), 61-68.
  • [17] L. V. Kantorovich, Concerning the general theory of operations in partially ordered spaces. Dok. Akad. Nauk. SSSR 1 (1936), 271-274.
  • [18] L. V. Kantorovich, Lineare halbgeordnete Raume. Rec. Math., 2 (1937), 121-168.
  • [19] Ş. Konca and E. Genc¸, Ideal version of weighted lacunary statitstical convergence for double sequences. Aligarh Bull. Math., 35(1-2) (2016), 83-97.
  • [20] Ş . Konca, E. Genc¸ and S. Ekin, Ideal version of weighted lacunary statistical convergence of sequences of order a. J. Math. Anal., 7(6) (2016), 19-30.
  • [21] Ş . Konca, Weighted lacunary I-statistical convergence. Igdır Uni. Fen Bilimleri Der./ Igdir Univ. J. Sci. Tech., 7(1) (2016), 267-277.
  • [22] P. Kostyrko, T. Salat and W. Wilczysnski, I-convergence. Real Anal. Exchange., 26(2) (2000-2001), 669-686.
  • [23] P. Kostyrko, M. Macaj, T. Salat, and M. Sleziak, I-convergence and extremal I-limit points. Math. Slovaca., 55(4) (2005), 443-464.
  • [24] P. Kostyrko, M. Macaj, T. Salat and O. Strauch, On statistical limit points. Proc. Amer. Math. Soc., 129(9) (2000), 2647-2654.
  • [25] P. Kostyrko, M. Macaj and T. Salat, Statistical convergence and I-convergence. to appear in Real Anal. Exchange.
  • [26] V. Kumar, On I and I-convergence of double sequences. Math. Commun., 12 (2007) 171-181.
  • [27] B. K. Lahiri and P. Das, I and I-convergence in topological spaces. Math. Bohem., 130(2) (2005), 153-160.
  • [28] W. A. Luxemburg and A. C. Zaanen, Riesz Spaces. American Elsevier Pub. Co., Vol. 1, (1971).
  • [29] S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces. Abst. Appl. Anal., Hindawi, 2012 Article ID 719729, (2012), 9 pages.
  • [30] A. Nabiev, S. Pehlivan and M. G¨urdal, On I-Cauchy sequence. Taiwanese J. Math., 11 (2) (2007), 569-576.
  • [31] F. Nuray, U. Ulusu and E. D¨undar, Lacunary statistical convergence of double sequences of sets. Soft Computing, 20(7) (2016), 2883-2888.
  • [32] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann., 53(3) (1900), 289-321.
  • [33] F. Riesz, Sur la decomposition des operations fonctionnelles lineaires, In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, (1929), 43-148.
  • [34] G. T. Roberts, Topologies in vector lattices, Math. Proc. Cambridge Phil. Soc., 48 (1952), 533-546.
  • [35] E. Savas¸, On I-lacunary double statistical convergence of weight g. Commun. Math. Appl., 8(2) (2017), 127-137.
  • [36] J. Schoenberg, The integrability of certain functions and related summability methods. Amer. Math. Monthly., 66 (1959), 361-375.
  • [37] H. Steinhaus, Sur la convergence ordinate et la convergence asymptotique. Colloq. Math., 2 (1951), 73-84.
  • [38] N. Subramanian and A. Es¸i, The backward operator of double almost (lmmn) convergence in c2-Riesz space defined by a Musielak-Orlicz, Bol. Soc. Paran. Mat. 37(3) (2019), 85-97.
  • [39] B. Tripathy, B. C. Tripathy, On I-convergent double sequences. Soochow J. Math., 31 (2005), 549-560.
  • [40] U. Ulusu and E. Dundar, I-Lacunary statistical convergence of sequences of sets. Filomat, 28(8) (2014), 1567-1574.
  • [41] U. Ulusu and F. Nuray, Lacunary statistical summability of sequences of sets. Konuralp J. Math., 3(2) (2015), 176-184.
  • [42] A. C. Zannen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, (1997).

Year 2019, Volume: 7 Issue: 1, 55 - 61, 15.04.2019

Şükran Konca , Ergin Genç Mehmet Küçükaslan

Abstract

References

  • [1] H. Albayrak, Statistical continuity and some convergence types in locally solid Riesz space. Phd Thesis, Institute of Science of S¨uleyman Demirel University, Isparta (2014).
  • [2] H. Albayrak and S. Pehlivan, Statistical convergence and statistical continuity on locally solid Riesz spaces. Top. Appl., 159(7) (2012), 1887-1893.
  • [3] C. D. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, Amer. Math. Soc. No. 105 (2003).
  • [4] A. Alotaibi, B. Hazarika and S. A. Mohiuddine, On the ideal convergence of double sequences in locally solid Riesz spaces, Abst. Appl. Anal., 2014 Hindawi, Article ID 396254, (2014), 6 pages.
  • [5] T. M. Apostol, Some properties of completely multiplicative arithmetical functions, Amer. Math. Monthly., 78(3) (1971), 266-271.
  • [6] M. Balcerzak, P. Das, M. Filipczak and J. Swaczyna, Generalized kinds of density and associated ideals. Acta Math. Hungar. 147(1) (2015), 97-115.
  • [7] P. Das, P. Kostyrko, W. Wilczyski and P. Malik, I and I-convergence of double sequences. Math. Slovaca, 58(5) (2008), 605-620.
  • [8] K. Dems, On I-Cauchy sequence, Real Anal. Exchange, 30 (2004/2005), 123-128.
  • [9] E. Dundar, U. Ulusu and B. Aydın, I2-lacunary statistical convergence of double sequences of sets. Konuralp J. Math. 5(1) (2016), 1-10.
  • [10] E. Dundar, B. Altay, On some properties of I2-convergence and I2-Cauchy of double sequences. Gen. Math. Notes, 7(1) (2011), 1-12.
  • [11] E. Dundar and B. Altay, I2-convergence and I2-Cauchy of double sequences. Acta Math. Sci., 34(2) (2014), 343-353.
  • [12] B. Hazarika and A. Eşi, On ideal convergence in locally solid Riesz spaces using lacunary mean. Proc. Jangjeon Math. Soc., 19(2) (2016), 253-262.
  • [13] B. Hazarika, A. Eşi, Quasi-Slowly oscillating sequences in locally normal Riesz spaces. Int. J. Anal. Appl., 15(2) (2017), 229-237. DOI: 10.28924/2291- 8639-15-2017-229.
  • [14] H. Fast, Sur la convergence statistique. Colloq. Math., 2(3-4) (1951), 241-244.
  • [15] H. Freudenthal, Teilweise geordnete Moduln, K. Akademie van Wetenschappen, Afdeeling Natuurkunde. Proc. Sec. Sci., 39 (1936), 647-657.
  • [16] B. Hazarika, S. A. Mohiuddine and M. Mursaleen, Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces. Iranian J. Sci. Tech., 38(1) (2014), 61-68.
  • [17] L. V. Kantorovich, Concerning the general theory of operations in partially ordered spaces. Dok. Akad. Nauk. SSSR 1 (1936), 271-274.
  • [18] L. V. Kantorovich, Lineare halbgeordnete Raume. Rec. Math., 2 (1937), 121-168.
  • [19] Ş. Konca and E. Genc¸, Ideal version of weighted lacunary statitstical convergence for double sequences. Aligarh Bull. Math., 35(1-2) (2016), 83-97.
  • [20] Ş . Konca, E. Genc¸ and S. Ekin, Ideal version of weighted lacunary statistical convergence of sequences of order a. J. Math. Anal., 7(6) (2016), 19-30.
  • [21] Ş . Konca, Weighted lacunary I-statistical convergence. Igdır Uni. Fen Bilimleri Der./ Igdir Univ. J. Sci. Tech., 7(1) (2016), 267-277.
  • [22] P. Kostyrko, T. Salat and W. Wilczysnski, I-convergence. Real Anal. Exchange., 26(2) (2000-2001), 669-686.
  • [23] P. Kostyrko, M. Macaj, T. Salat, and M. Sleziak, I-convergence and extremal I-limit points. Math. Slovaca., 55(4) (2005), 443-464.
  • [24] P. Kostyrko, M. Macaj, T. Salat and O. Strauch, On statistical limit points. Proc. Amer. Math. Soc., 129(9) (2000), 2647-2654.
  • [25] P. Kostyrko, M. Macaj and T. Salat, Statistical convergence and I-convergence. to appear in Real Anal. Exchange.
  • [26] V. Kumar, On I and I-convergence of double sequences. Math. Commun., 12 (2007) 171-181.
  • [27] B. K. Lahiri and P. Das, I and I-convergence in topological spaces. Math. Bohem., 130(2) (2005), 153-160.
  • [28] W. A. Luxemburg and A. C. Zaanen, Riesz Spaces. American Elsevier Pub. Co., Vol. 1, (1971).
  • [29] S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces. Abst. Appl. Anal., Hindawi, 2012 Article ID 719729, (2012), 9 pages.
  • [30] A. Nabiev, S. Pehlivan and M. G¨urdal, On I-Cauchy sequence. Taiwanese J. Math., 11 (2) (2007), 569-576.
  • [31] F. Nuray, U. Ulusu and E. D¨undar, Lacunary statistical convergence of double sequences of sets. Soft Computing, 20(7) (2016), 2883-2888.
  • [32] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann., 53(3) (1900), 289-321.
  • [33] F. Riesz, Sur la decomposition des operations fonctionnelles lineaires, In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, (1929), 43-148.
  • [34] G. T. Roberts, Topologies in vector lattices, Math. Proc. Cambridge Phil. Soc., 48 (1952), 533-546.
  • [35] E. Savas¸, On I-lacunary double statistical convergence of weight g. Commun. Math. Appl., 8(2) (2017), 127-137.
  • [36] J. Schoenberg, The integrability of certain functions and related summability methods. Amer. Math. Monthly., 66 (1959), 361-375.
  • [37] H. Steinhaus, Sur la convergence ordinate et la convergence asymptotique. Colloq. Math., 2 (1951), 73-84.
  • [38] N. Subramanian and A. Es¸i, The backward operator of double almost (lmmn) convergence in c2-Riesz space defined by a Musielak-Orlicz, Bol. Soc. Paran. Mat. 37(3) (2019), 85-97.
  • [39] B. Tripathy, B. C. Tripathy, On I-convergent double sequences. Soochow J. Math., 31 (2005), 549-560.
  • [40] U. Ulusu and E. Dundar, I-Lacunary statistical convergence of sequences of sets. Filomat, 28(8) (2014), 1567-1574.
  • [41] U. Ulusu and F. Nuray, Lacunary statistical summability of sequences of sets. Konuralp J. Math., 3(2) (2015), 176-184.
  • [42] A. C. Zannen, Introduction to Operator Theory in Riesz Spaces, Springer-Verlag, (1997).
There are 42 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Şükran Konca

Ergin Genç This is me

Mehmet Küçükaslan

Publication Date April 15, 2019
Submission Date December 7, 2018
Acceptance Date February 14, 2019
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Konca, Ş., Genç, E., & Küçükaslan, M. (2019). I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space. Konuralp Journal of Mathematics, 7(1), 55-61.
AMA Konca Ş, Genç E, Küçükaslan M. I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space. Konuralp J. Math. April 2019;7(1):55-61.
Chicago Konca, Şükran, Ergin Genç, and Mehmet Küçükaslan. “I-Statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space”. Konuralp Journal of Mathematics 7, no. 1 (April 2019): 55-61.
EndNote Konca Ş, Genç E, Küçükaslan M (April 1, 2019) I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space. Konuralp Journal of Mathematics 7 1 55–61.
IEEE Ş. Konca, E. Genç, and M. Küçükaslan, “I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space”, Konuralp J. Math., vol. 7, no. 1, pp. 55–61, 2019.
ISNAD Konca, Şükran et al. “I-Statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space”. Konuralp Journal of Mathematics 7/1 (April 2019), 55-61.
JAMA Konca Ş, Genç E, Küçükaslan M. I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space. Konuralp J. Math. 2019;7:55–61.
MLA Konca, Şükran et al. “I-Statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space”. Konuralp Journal of Mathematics, vol. 7, no. 1, 2019, pp. 55-61.
Vancouver Konca Ş, Genç E, Küçükaslan M. I-statistical Convergence of Double Sequences Defined by Weight Functions in a Locally Solid Riesz Space. Konuralp J. Math. 2019;7(1):55-61.
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