Research Article
BibTex RIS Cite

On strong N_{θ}^{α}(A,F)-convergence

Year 2019, Volume: 68 Issue: 2, 1629 - 1637, 01.08.2019
https://doi.org/10.31801/cfsuasmas.546616

Abstract

In the papers [T. Bilgin, Studia Univ. Babeş-Bolyai Math. 46(4), (2001), 39--46] and [T. Bilgin, Appl. Math. Comput. 151(3), (2004), 595--600], author defined the spaces of strongly N_{θ}(A,f)-convergent with respect to a modulus sequences and strongly N_{θ}(A,F)-convergent with respect to a sequence of modulus functions sequences. In this paper, we introduce strong N_{θ}^{α}(A,F)-convergence with respect to a sequence of modulus functions and give some connections between sets of strongly N_{θ}^{α}(A,F)-convergent with respect to a sequence of modulus functions sequences and S_{θ}^{α}(A)-convergent sequences.

References

  • Bilgin, T., Lacunary strong A-convergence with respect to a modulus, Studia Univ. Babeş-Bolyai Math., 46(4) (2001) 39--46.
  • Bilgin, T., Lacunary strong A-convergence with respect to a sequence of modulus functions, Appl. Math. Comput., 151(3) (2004) 595--600.
  • Altin, Y. and Et, M., Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow J. Math., 31(2) (2005) 233--243.
  • Bhardwaj, V. K. and Dhawan, S., Density by moduli and lacunary statistical convergence, Abstr. Appl. Anal., 2016 (2016), Art. ID 9365037, 11 pp.
  • Caserta, A., Di Maio, G. and Kočinac, L. D. R., Statistical convergence in function spaces, Abstr. Appl. Anal., 2011 (2011), Art. ID 420419, 11 pp.
  • Connor, J. S., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988) 47--63.
  • Demirci, K., Strong A-summability and A-statistical convergence, Indian J. Pure Appl. Math., 27(6) (1996) 589--593.
  • Cakalli, H., Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math., 26(2) (1995) 113--119.
  • Cakalli, H., A study on statistical convergence, Funct. Anal. Approx. Comput., 1(2) (2009) 19--24.
  • Cinar, M., Karakas, M. and Et, M., On pointwise and uniform statistical convergence of order α for sequences of functions, Fixed Point Theory And Applications, 2013(33) (2013) 11 pp.
  • Colak, R., Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub, 2010 (2010) 121--129.
  • Di Maio, G. and Kočinac, L. D. R., Statistical convergence in topology, Topology Appl., 156 (2008) 28--45.
  • Et, M., Cinar, M. and Karakas, M., On λ-statistical convergence of order α of sequences of functions, J. Inequal. Appl., 2013(204) (2013) 8 pp.
  • Et, M., Altin, Y. and Altinok, H., On some generalized difference sequence spaces defined by a modulus function, Filomat, 17 (2003) 23--33.
  • Et, M., Alotaibi, A. and Mohiuddine, S. A., On (Δ^{m},I) statistical convergence of order α, Scientific World Journal, 2014 (2014) Article Number: 535419.
  • Et, M. and Sengul, H., Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28(8) (2014) 1593--1602.
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951) 241--244.
  • Freedman, A. R., Sember, J. J. and Raphael, M., Some Cesàro-type summability spaces, Proc. London Math. Soc., 37(3) (1978) 508--520.
  • Fridy, J., On statistical convergence, Analysis, 5 (1985) 301--313.
  • Fridy, J. and Orhan, C., Lacunary statistical convergence, Pacific J. Math., 160 (1993) 43--51.
  • Gaur, A. K. and Mursaleen, M., Difference sequence spaces defined by a sequence of moduli, Demonstratio Math., 31(2) (1998) 275--278.
  • Isik, M. and Et, K. E., On lacunary statistical convergence of order α in probability, AIP Conference Proceedings, 1676, 020045 (2015); doi: http://dx.doi.org/10.1063/1.4930471.
  • Isik, M. and Akbas, K. E., On λ-statistical convergence of order α in probability, J. Inequal. Spec. Funct., 8(4) (2017) 57--64.
  • Isik, M., Generalized vector-valued sequence spaces defined by modulus functions, J. Inequal. Appl., 2010 (2010) Art. ID 457892, 7 pp.
  • Kaplan, H. and Cakalli, H., Variations on strong lacunary quasi-Cauchy sequences, J. Nonlinear Sci. Appl., 9(6) (2016) 4371--4380.
  • Maddox, I. J., Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc., 100 (1986) 161--166.
  • Nakano, H., Modulared sequence spaces, Proc. Japan Acad., 27 (1951) 508--512.
  • Nuray, F. and Savas, E., Some new sequence spaces defined by a modulus function, Indian J. Pure Appl. Math., 24(11) (1993) 657--663.
  • Pehlivan, S. and Fisher, B., Lacunary strong convergence with respect to a sequence of modulus functions, Comment. Math. Univ. Carolin., 36(1) (1995) 69--76.
  • Pehlivan, S. and Fisher, B., Some sequence spaces defined by a modulus, Math. Slovaca. 45(3) (1995) 275--280.
  • Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980) 139--150.
  • Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959) 361--375.
  • Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951) 73--74.
  • Sengul, H., Some Cesàro-type summability spaces defined by a modulus function of order (α,β), Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 66(2) (2017) 80--90.
  • Sengul, H. and Et, M., On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014) 473--482.
  • Sengul, H. and Arica, Z., Lacunary A-statistical convergence and lacunary strong A-convergence of order α with respect to a modulus, Conference Proceedings of ICMS-18, (2018), Maltepe/ Istanbul.
  • Altinok, H. and Yagdiran, D., Lacunary statistical convergence defined by an Orlicz function in sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems, 32(3) (2017) 2725--2731.
  • Altinok, H. and Yagdiran, D., Lacunary statistical convergence of order β in difference sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems, 31(1) (2016) 227--235.
Year 2019, Volume: 68 Issue: 2, 1629 - 1637, 01.08.2019
https://doi.org/10.31801/cfsuasmas.546616

Abstract

References

  • Bilgin, T., Lacunary strong A-convergence with respect to a modulus, Studia Univ. Babeş-Bolyai Math., 46(4) (2001) 39--46.
  • Bilgin, T., Lacunary strong A-convergence with respect to a sequence of modulus functions, Appl. Math. Comput., 151(3) (2004) 595--600.
  • Altin, Y. and Et, M., Generalized difference sequence spaces defined by a modulus function in a locally convex space, Soochow J. Math., 31(2) (2005) 233--243.
  • Bhardwaj, V. K. and Dhawan, S., Density by moduli and lacunary statistical convergence, Abstr. Appl. Anal., 2016 (2016), Art. ID 9365037, 11 pp.
  • Caserta, A., Di Maio, G. and Kočinac, L. D. R., Statistical convergence in function spaces, Abstr. Appl. Anal., 2011 (2011), Art. ID 420419, 11 pp.
  • Connor, J. S., The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988) 47--63.
  • Demirci, K., Strong A-summability and A-statistical convergence, Indian J. Pure Appl. Math., 27(6) (1996) 589--593.
  • Cakalli, H., Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math., 26(2) (1995) 113--119.
  • Cakalli, H., A study on statistical convergence, Funct. Anal. Approx. Comput., 1(2) (2009) 19--24.
  • Cinar, M., Karakas, M. and Et, M., On pointwise and uniform statistical convergence of order α for sequences of functions, Fixed Point Theory And Applications, 2013(33) (2013) 11 pp.
  • Colak, R., Statistical convergence of order α, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub, 2010 (2010) 121--129.
  • Di Maio, G. and Kočinac, L. D. R., Statistical convergence in topology, Topology Appl., 156 (2008) 28--45.
  • Et, M., Cinar, M. and Karakas, M., On λ-statistical convergence of order α of sequences of functions, J. Inequal. Appl., 2013(204) (2013) 8 pp.
  • Et, M., Altin, Y. and Altinok, H., On some generalized difference sequence spaces defined by a modulus function, Filomat, 17 (2003) 23--33.
  • Et, M., Alotaibi, A. and Mohiuddine, S. A., On (Δ^{m},I) statistical convergence of order α, Scientific World Journal, 2014 (2014) Article Number: 535419.
  • Et, M. and Sengul, H., Some Cesaro-type summability spaces of order α and lacunary statistical convergence of order α, Filomat, 28(8) (2014) 1593--1602.
  • Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951) 241--244.
  • Freedman, A. R., Sember, J. J. and Raphael, M., Some Cesàro-type summability spaces, Proc. London Math. Soc., 37(3) (1978) 508--520.
  • Fridy, J., On statistical convergence, Analysis, 5 (1985) 301--313.
  • Fridy, J. and Orhan, C., Lacunary statistical convergence, Pacific J. Math., 160 (1993) 43--51.
  • Gaur, A. K. and Mursaleen, M., Difference sequence spaces defined by a sequence of moduli, Demonstratio Math., 31(2) (1998) 275--278.
  • Isik, M. and Et, K. E., On lacunary statistical convergence of order α in probability, AIP Conference Proceedings, 1676, 020045 (2015); doi: http://dx.doi.org/10.1063/1.4930471.
  • Isik, M. and Akbas, K. E., On λ-statistical convergence of order α in probability, J. Inequal. Spec. Funct., 8(4) (2017) 57--64.
  • Isik, M., Generalized vector-valued sequence spaces defined by modulus functions, J. Inequal. Appl., 2010 (2010) Art. ID 457892, 7 pp.
  • Kaplan, H. and Cakalli, H., Variations on strong lacunary quasi-Cauchy sequences, J. Nonlinear Sci. Appl., 9(6) (2016) 4371--4380.
  • Maddox, I. J., Sequence spaces defined by a modulus, Math. Proc. Camb. Philos. Soc., 100 (1986) 161--166.
  • Nakano, H., Modulared sequence spaces, Proc. Japan Acad., 27 (1951) 508--512.
  • Nuray, F. and Savas, E., Some new sequence spaces defined by a modulus function, Indian J. Pure Appl. Math., 24(11) (1993) 657--663.
  • Pehlivan, S. and Fisher, B., Lacunary strong convergence with respect to a sequence of modulus functions, Comment. Math. Univ. Carolin., 36(1) (1995) 69--76.
  • Pehlivan, S. and Fisher, B., Some sequence spaces defined by a modulus, Math. Slovaca. 45(3) (1995) 275--280.
  • Salat, T., On statistically convergent sequences of real numbers, Math. Slovaca, 30 (1980) 139--150.
  • Schoenberg, I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959) 361--375.
  • Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951) 73--74.
  • Sengul, H., Some Cesàro-type summability spaces defined by a modulus function of order (α,β), Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 66(2) (2017) 80--90.
  • Sengul, H. and Et, M., On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed. 34(2) (2014) 473--482.
  • Sengul, H. and Arica, Z., Lacunary A-statistical convergence and lacunary strong A-convergence of order α with respect to a modulus, Conference Proceedings of ICMS-18, (2018), Maltepe/ Istanbul.
  • Altinok, H. and Yagdiran, D., Lacunary statistical convergence defined by an Orlicz function in sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems, 32(3) (2017) 2725--2731.
  • Altinok, H. and Yagdiran, D., Lacunary statistical convergence of order β in difference sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems, 31(1) (2016) 227--235.
There are 38 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Hacer Şengül 0000-0003-4453-0786

Zelal Arıca This is me

Publication Date August 1, 2019
Submission Date July 31, 2018
Acceptance Date January 15, 2019
Published in Issue Year 2019 Volume: 68 Issue: 2

Cite

APA Şengül, H., & Arıca, Z. (2019). On strong N_{θ}^{α}(A,F)-convergence. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1629-1637. https://doi.org/10.31801/cfsuasmas.546616
AMA Şengül H, Arıca Z. On strong N_{θ}^{α}(A,F)-convergence. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. August 2019;68(2):1629-1637. doi:10.31801/cfsuasmas.546616
Chicago Şengül, Hacer, and Zelal Arıca. “On Strong N_{θ}^{α}(A,F)-Convergence”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 2 (August 2019): 1629-37. https://doi.org/10.31801/cfsuasmas.546616.
EndNote Şengül H, Arıca Z (August 1, 2019) On strong N_{θ}^{α}(A,F)-convergence. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 2 1629–1637.
IEEE H. Şengül and Z. Arıca, “On strong N_{θ}^{α}(A,F)-convergence”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 2, pp. 1629–1637, 2019, doi: 10.31801/cfsuasmas.546616.
ISNAD Şengül, Hacer - Arıca, Zelal. “On Strong N_{θ}^{α}(A,F)-Convergence”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/2 (August 2019), 1629-1637. https://doi.org/10.31801/cfsuasmas.546616.
JAMA Şengül H, Arıca Z. On strong N_{θ}^{α}(A,F)-convergence. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1629–1637.
MLA Şengül, Hacer and Zelal Arıca. “On Strong N_{θ}^{α}(A,F)-Convergence”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 2, 2019, pp. 1629-37, doi:10.31801/cfsuasmas.546616.
Vancouver Şengül H, Arıca Z. On strong N_{θ}^{α}(A,F)-convergence. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(2):1629-37.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.