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ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS

Yıl 2020, Cilt: 2 Sayı: 1, 1 - 8, 30.04.2020

Bayram Sözbir Selma Altundağ Metin Basarır

Öz

In this paper, we introduce the concept of ∆f -lacunary statistical convergence for a ∆-measurable real-valued function defined on time scale, where f is an unbounded modulus. Our motivation here is that this definition includes many well-known concepts which already exist in the literature. We also define strong ∆f -lacunary Cesaro summability on a time scale and give some results related to these new concepts. Furthermore, we obtain necessary and sufficient conditions for the equivalence of ∆f-convergence and ∆f -lacunary statistical convergence on a time scale.

Anahtar Kelimeler

Modulus function lacunary statistical convergence delta measure on time scale time scale

Kaynakça

  • [1] Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241-244 (1951).
  • [2] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1), 73-74 (1951).
  • [3] Schoenberg, I.J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66, 361-375 (1959).
  • [4] Fridy, J.A.: On statistical convergence. Analysis 5, 301-313 (1985).
  • [5] Fridy, J.A., Orhan, C.: Lacunary statistical convergence. Paci c J. Math. 160, 43-51 (1993).
  • [6] Connor, J.S.: The statistical and strong p-Cesaro convergence of sequences. Analysis 8, 47-63 (1988).
  • [7] Connor, J.S.: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 32, 194-198 (1989).
  • [8] Moricz, F.: Statistical limits of measurable functions. Analysis 24 (1), 1-18 (2004).
  • [9] Nakano, H.: Concave modulars. J. Math. Soc. Japan. 5, 29-49 (1953).
  • [10] Ruckle, W.H.: FK spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 25, 973-978 (1973).
  • [11] Maddox, I.J.: Sequence spaces defi ned by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1), 161-166 (1986).
  • [12] Maddox, I.J.: Inclusions between FK spaces and Kuttner's theorem. Math. Proc. Cambridge Philos. Soc. 101 (3), 523-527 (1987).
  • [13] Aizpuru, A., Listan-Garcia, M.C., Rambla-Barreno, F.: Density by moduli and statistical convergence. Quaestiones Mathematicae, 37 (4), 525-530 (2014).
  • [14] Bhardwaj, V.K., Dhawan, S.: Density by moduli and lacunary statistical convergence. Abst. Appl. Analysis. 2016, (2016).
  • [15] Hilger, S.: Analysis on measure chains-a uni ed approach to continuous and discrete calculus. Results Math. 18 (1-2), 18-56 (1990).
  • [16] Bohner, M., Peterson, A.: Dynamic equations on time scales: An introduction with applications, Birkhauser, Boston (2001).
  • [17] Agarwal, R., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4 (4), 535-558 (2001).
  • [18] Guseinov, G.S.: Integration on time scales. J. Math. Anal. Appl. 285 (1), 107-127 (2003).
  • [19] Cabada, A., Vivero, D.R.: Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral: Application to the calculus of ∆-antiderivatives. Math. Comput. Model. 43 (1-2), 194-207 (2006).
  • [20] Seyyidoglu, M.S., Tan, N.O.: A note on statistical convergence on time scale. J. Inequal. Appl. 2012 (219), (2012).
  • [21] Turan, C., Duman, O.: Statistical convergence on time scales and its characterizations. Springer Proc. Math. Stat. 41, 57-71 (2013).
  • [22] Turan, C., Duman, O.: Convergence methods on time scales. AIP Conf. Proc. 1558 (1), 1120-1123 (2013).
  • [23] Turan, C., Duman, O.: Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 31 (14), 4455-4467 (2017).
  • [24] Altin, Y., Koyunbakan, H., Yilmaz, E.: Uniform statistical convergence on time scales. J. Appl. Math. 2014, (2014).
  • [25] Sozbir, B., Altundag, S.: Weighted statistical convergence on time scale. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 26, 137-143 (2019).
  • [26] Turan, N., Basarir, M.: On the ∆g-statistical convergence of the function de fined time scale, AIP Conference Proceedings, 2183, 040017 (2019). https://doi.org/10.1063/1.5136137.

Yıl 2020, Cilt: 2 Sayı: 1, 1 - 8, 30.04.2020

Bayram Sözbir Selma Altundağ Metin Basarır

Öz

Kaynakça

  • [1] Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241-244 (1951).
  • [2] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1), 73-74 (1951).
  • [3] Schoenberg, I.J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66, 361-375 (1959).
  • [4] Fridy, J.A.: On statistical convergence. Analysis 5, 301-313 (1985).
  • [5] Fridy, J.A., Orhan, C.: Lacunary statistical convergence. Paci c J. Math. 160, 43-51 (1993).
  • [6] Connor, J.S.: The statistical and strong p-Cesaro convergence of sequences. Analysis 8, 47-63 (1988).
  • [7] Connor, J.S.: On strong matrix summability with respect to a modulus and statistical convergence. Canad. Math. Bull. 32, 194-198 (1989).
  • [8] Moricz, F.: Statistical limits of measurable functions. Analysis 24 (1), 1-18 (2004).
  • [9] Nakano, H.: Concave modulars. J. Math. Soc. Japan. 5, 29-49 (1953).
  • [10] Ruckle, W.H.: FK spaces in which the sequence of coordinate vectors is bounded. Can. J. Math. 25, 973-978 (1973).
  • [11] Maddox, I.J.: Sequence spaces defi ned by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1), 161-166 (1986).
  • [12] Maddox, I.J.: Inclusions between FK spaces and Kuttner's theorem. Math. Proc. Cambridge Philos. Soc. 101 (3), 523-527 (1987).
  • [13] Aizpuru, A., Listan-Garcia, M.C., Rambla-Barreno, F.: Density by moduli and statistical convergence. Quaestiones Mathematicae, 37 (4), 525-530 (2014).
  • [14] Bhardwaj, V.K., Dhawan, S.: Density by moduli and lacunary statistical convergence. Abst. Appl. Analysis. 2016, (2016).
  • [15] Hilger, S.: Analysis on measure chains-a uni ed approach to continuous and discrete calculus. Results Math. 18 (1-2), 18-56 (1990).
  • [16] Bohner, M., Peterson, A.: Dynamic equations on time scales: An introduction with applications, Birkhauser, Boston (2001).
  • [17] Agarwal, R., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4 (4), 535-558 (2001).
  • [18] Guseinov, G.S.: Integration on time scales. J. Math. Anal. Appl. 285 (1), 107-127 (2003).
  • [19] Cabada, A., Vivero, D.R.: Expression of the Lebesgue ∆-integral on time scales as a usual Lebesgue integral: Application to the calculus of ∆-antiderivatives. Math. Comput. Model. 43 (1-2), 194-207 (2006).
  • [20] Seyyidoglu, M.S., Tan, N.O.: A note on statistical convergence on time scale. J. Inequal. Appl. 2012 (219), (2012).
  • [21] Turan, C., Duman, O.: Statistical convergence on time scales and its characterizations. Springer Proc. Math. Stat. 41, 57-71 (2013).
  • [22] Turan, C., Duman, O.: Convergence methods on time scales. AIP Conf. Proc. 1558 (1), 1120-1123 (2013).
  • [23] Turan, C., Duman, O.: Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 31 (14), 4455-4467 (2017).
  • [24] Altin, Y., Koyunbakan, H., Yilmaz, E.: Uniform statistical convergence on time scales. J. Appl. Math. 2014, (2014).
  • [25] Sozbir, B., Altundag, S.: Weighted statistical convergence on time scale. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 26, 137-143 (2019).
  • [26] Turan, N., Basarir, M.: On the ∆g-statistical convergence of the function de fined time scale, AIP Conference Proceedings, 2183, 040017 (2019). https://doi.org/10.1063/1.5136137.
Toplam 26 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Bayram Sözbir 0000-0002-9475-7180

Selma Altundağ 0000-0002-5893-9868

Metin Basarır 0000-0002-4341-4399

Yayımlanma Tarihi 30 Nisan 2020
Kabul Tarihi 28 Nisan 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 2 Sayı: 1

Kaynak Göster

APA Sözbir, B., Altundağ, S., & Basarır, M. (2020). ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS. Maltepe Journal of Mathematics, 2(1), 1-8.
AMA Sözbir B, Altundağ S, Basarır M. ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS. Maltepe Journal of Mathematics. Nisan 2020;2(1):1-8.
Chicago Sözbir, Bayram, Selma Altundağ, ve Metin Basarır. “ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS”. Maltepe Journal of Mathematics 2, sy. 1 (Nisan 2020): 1-8.
EndNote Sözbir B, Altundağ S, Basarır M (01 Nisan 2020) ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS. Maltepe Journal of Mathematics 2 1 1–8.
IEEE B. Sözbir, S. Altundağ, ve M. Basarır, “ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS”, Maltepe Journal of Mathematics, c. 2, sy. 1, ss. 1–8, 2020.
ISNAD Sözbir, Bayram vd. “ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS”. Maltepe Journal of Mathematics 2/1 (Nisan 2020), 1-8.
JAMA Sözbir B, Altundağ S, Basarır M. ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS. Maltepe Journal of Mathematics. 2020;2:1–8.
MLA Sözbir, Bayram vd. “ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS”. Maltepe Journal of Mathematics, c. 2, sy. 1, 2020, ss. 1-8.
Vancouver Sözbir B, Altundağ S, Basarır M. ON THE (DELTA,f)-LACUNARY STATISTICAL CONVERGENCE OF THE FUNCTIONS. Maltepe Journal of Mathematics. 2020;2(1):1-8.

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