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

Year 2020, Volume: 2 Issue: 1, 1 - 8, 30.04.2020

Bayram Sözbir , Selma Altundağ , Metin Basarır

https://izlik.org/JA84ZZ34DD

Abstract

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.

Keywords

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

References

• [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.