On strong N_{θ}^{α}(A,F)-convergence

Yıl 2019, Cilt: 68 Sayı: 2, 1629 - 1637, 01.08.2019

Hacer Şengül , Zelal Arıca

https://doi.org/10.31801/cfsuasmas.546616

Öz

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.

Anahtar Kelimeler

Modulus function, statistical convergence, lacunary sequence

Kaynakça

• 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.