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.
Keywords
Lacunary sequence I-convergence Invariant convergence Statistical convergence I-Cauchy sequence
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.
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.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Natural Sciences |
| Authors | |
| Project Number | 16.KARİYER.62 |
| Publication Date | September 30, 2020 |
| Submission Date | February 16, 2020 |
| Acceptance Date | June 26, 2020 |
| Published in Issue | Year 2020Volume: 41 Issue: 3 |