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Lacunary I-invariant convergence

Year 2020, , 617 - 624, 30.09.2020
https://doi.org/10.17776/csj.689877

Abstract

In this study, firstly, we introduce the notion of lacunary invariant uniform density of any subset E of the set N (the set of all natural numbers). Then, as associated with this notion, we give the definition of lacunary I_σ-convergence for real sequences. Furthermore, we examine relations between this new type convergence notion and the notions of lacunary invariant summability, lacunary strongly q-invariant summability and lacunary σ-statistical convergence which are studied in this area before. Finally, introducing the notions of lacunary I_σ^*-convergence and I_σ-Cauchy sequence, we give the relations between these notions and the notion of lacunary I_σ-convergence.

Supporting Institution

The Scientific Research Project Fund of Afyon Kocatepe University

Project Number

16.KARİYER.62

Thanks

We would like to thank "The Scientific Research Project Fund of Afyon Kocatepe University" for its support with this project, project number 16.KARİYER.62.

References

  • Raimi R.A., Invariant means and invariant matrix methods of summability, Duke Math. J., 30(1) (1963) 81-94.
  • Schaefer P., Infinite matrices and invariant means. Proc. Amer. Math. Soc., 36(1) (1972) 104-110.
  • Mursaleen M., On finite matrices and invariant means, Indian J. Pure Appl. Math., 10 (1979) 457-460.
  • Savaş E., Strongly σ-convergent sequences, Bull. Calcutta Math., 81 (1989) 295-300.
  • Mursaleen M. and Edely O.H.H., On the invariant mean and statistical convergenc, Appl. Math. Lett., 22(11) (2009) 1700-1704.
  • Başarır M. and Konca Ş., On some lacunary almost convergent double sequence spaces and Banach limits, Abstr. Appl. Anal., 2012 (2012).
  • Pancaroğlu N. and Nuray F., On invariant statistically convergence and lacunary invariant statistical convergence of sequences of sets, Prog. Appl. Math., 5(2) (2013) 23-29.
  • Nuray F. and Ulusu U., Lacunary invariant statistical convergence of double sequences of sets, Creat. Math. Inform., 28(2) (2019) 143-150.
  • Mursaleen M., Matrix transformation between some new sequence spaces, Houston J. Math., 9 (1983) 505-509.
  • Savaş E., Some sequence spaces involving invariant means, Indian J. Math., 31 (1989) 1-8.
  • Fridy J.A.,and Orhan C., Lacunary statistical convergence, Pacific J. Math., 160(1) (1993) 43-51.
  • Savaş E., On lacunary strong σ-convergence, Indian J. Pure Appl. Math., 21(4) (1990) 359-365.
  • Pancaroğlu N. and Nuray F., Statistical lacunary invariant summability, Theoretical Math. Appl., 3(2) (2013) 71-78.
  • Fast H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951) 241-244.
  • Šalát T., On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980) 139-150.
  • Fridy J.A., On statistical convergence, Analysis, 5(4) (1985) 301-314.
  • Rath D. and Tripathy B.C., On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25(4) (1994) 381-386.
  • Savaş E. and Nuray F., On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca, 43(3) (1993) 309-315.
  • Kostyrko P., Šalát T. and Wilczyński W., I-convergence, Real Anal. Exchange, 26(2) (2000) 669-686.
  • Kostyrko P., Macaj M., Šalát T., Sleziak M., I-convergence and external I-limits points, Math. Slovaca, 55 (2005) 443-464.
  • Sever Y., Ulusu U. and Dündar E., On strongly I and I^*-lacunary convergence of sequences of sets, AIP Conf. Proc., 1611(1) (2014) 357–362.
  • Konca Ş., Weighted lacunary I-statistical convergence, Iğdır Univ. J. Inst. Sci. & Tech., 7(1) (2017) 267-277.
  • Nabiev A., Pehlivan S. and Gürdal M., On I-Cauchy sequences, Taiwanese J. Math., 11(2) (2007) 569-576.
  • Dems K., On I-Cauchy sequences, Real Anal. Exchange, 30(1) (2004) 123-128.
  • Nuray F., Gök H., Ulusu U., I_σ-convergence, Math Commun., 16 (2011) 531-538.
Year 2020, , 617 - 624, 30.09.2020
https://doi.org/10.17776/csj.689877

Abstract

Project Number

16.KARİYER.62

References

  • Raimi R.A., Invariant means and invariant matrix methods of summability, Duke Math. J., 30(1) (1963) 81-94.
  • Schaefer P., Infinite matrices and invariant means. Proc. Amer. Math. Soc., 36(1) (1972) 104-110.
  • Mursaleen M., On finite matrices and invariant means, Indian J. Pure Appl. Math., 10 (1979) 457-460.
  • Savaş E., Strongly σ-convergent sequences, Bull. Calcutta Math., 81 (1989) 295-300.
  • Mursaleen M. and Edely O.H.H., On the invariant mean and statistical convergenc, Appl. Math. Lett., 22(11) (2009) 1700-1704.
  • Başarır M. and Konca Ş., On some lacunary almost convergent double sequence spaces and Banach limits, Abstr. Appl. Anal., 2012 (2012).
  • Pancaroğlu N. and Nuray F., On invariant statistically convergence and lacunary invariant statistical convergence of sequences of sets, Prog. Appl. Math., 5(2) (2013) 23-29.
  • Nuray F. and Ulusu U., Lacunary invariant statistical convergence of double sequences of sets, Creat. Math. Inform., 28(2) (2019) 143-150.
  • Mursaleen M., Matrix transformation between some new sequence spaces, Houston J. Math., 9 (1983) 505-509.
  • Savaş E., Some sequence spaces involving invariant means, Indian J. Math., 31 (1989) 1-8.
  • Fridy J.A.,and Orhan C., Lacunary statistical convergence, Pacific J. Math., 160(1) (1993) 43-51.
  • Savaş E., On lacunary strong σ-convergence, Indian J. Pure Appl. Math., 21(4) (1990) 359-365.
  • Pancaroğlu N. and Nuray F., Statistical lacunary invariant summability, Theoretical Math. Appl., 3(2) (2013) 71-78.
  • Fast H., Sur la convergence statistique, Colloq. Math., 2(3-4) (1951) 241-244.
  • Šalát T., On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980) 139-150.
  • Fridy J.A., On statistical convergence, Analysis, 5(4) (1985) 301-314.
  • Rath D. and Tripathy B.C., On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25(4) (1994) 381-386.
  • Savaş E. and Nuray F., On σ-statistically convergence and lacunary σ-statistically convergence, Math. Slovaca, 43(3) (1993) 309-315.
  • Kostyrko P., Šalát T. and Wilczyński W., I-convergence, Real Anal. Exchange, 26(2) (2000) 669-686.
  • Kostyrko P., Macaj M., Šalát T., Sleziak M., I-convergence and external I-limits points, Math. Slovaca, 55 (2005) 443-464.
  • Sever Y., Ulusu U. and Dündar E., On strongly I and I^*-lacunary convergence of sequences of sets, AIP Conf. Proc., 1611(1) (2014) 357–362.
  • Konca Ş., Weighted lacunary I-statistical convergence, Iğdır Univ. J. Inst. Sci. & Tech., 7(1) (2017) 267-277.
  • Nabiev A., Pehlivan S. and Gürdal M., On I-Cauchy sequences, Taiwanese J. Math., 11(2) (2007) 569-576.
  • Dems K., On I-Cauchy sequences, Real Anal. Exchange, 30(1) (2004) 123-128.
  • Nuray F., Gök H., Ulusu U., I_σ-convergence, Math Commun., 16 (2011) 531-538.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Uğur Ulusu 0000-0001-7658-6114

Fatih Nuray 0000-0003-0160-4001

Project Number 16.KARİYER.62
Publication Date September 30, 2020
Submission Date February 16, 2020
Acceptance Date June 26, 2020
Published in Issue Year 2020

Cite

APA Ulusu, U., & Nuray, F. (2020). Lacunary I-invariant convergence. Cumhuriyet Science Journal, 41(3), 617-624. https://doi.org/10.17776/csj.689877