On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale

Year 2020, Volume: 3 Issue: 4, 138 - 143, 23.12.2020

Bayram Sözbir , Selma Altundağ , Metin Basarır

https://doi.org/10.32323/ujma.743949

Cited By: 1

Abstract

In this paper, we first define a new density of a $\Delta $-measurable subset of a product time scale ${\Lambda ^2}$ with respect to an unbounded modulus function. Then, by using this definition, we introduce the concepts of $\Delta _{{\Lambda ^2}}^f$-statistical convergence and $\Delta _{{\Lambda ^2}}^f$-statistical Cauchy for a $\Delta $-measurable real-valued function defined on product time scale ${\Lambda ^2}$ and also obtain some results about these new concepts. Finally, we present the definition of strong $\Delta _{{\Lambda ^2}}^f$-Cesaro summability on ${\Lambda ^2}$ and investigate the connections between these new concepts.

Keywords

Delta measure, Density, Modulus function, Product time scale, Statistical convergence, Strong Cesaro summability

References

• [1] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
• [2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2(1) (1951), 73–74.
• [3] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
• [4] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
• [5] J.S. Connor, The statistical and strong p-Cesaro convergence of sequences, Analysis, 8 (1988), 47–63.
• [6] M. Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl., 288(1) (2003), 223–231.
• [7] F. Moricz, Statistical limits of measurable functions, Analysis, 24(1) (2004), 1–18.
• [8] E. D¨undar, Y. Sever, Multipliers for bounded statistical convergence of double Sequences, Int. Math. Forum, 7(52) (2012), 2581–2587.
• [9] U. Ulusu, E. Dündar, I-lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567–1574, DOI 10.2298/FIL1408567U.
• [10] F. Nuray, U. Ulusu, E. Dündar, Lacunary statistical convergence of double sequences of sets, Soft Comput., 20 (2016), 2883–2888, DOI 10.1007/s00500- 015-1691-8.
• [11] S. Yegül, E. Dündar, On statistical convergence of sequences of functions in 2-normed spaces, J. Classical Anal., 10(1) (2017), 49–57.
• [12] S. Yegül, E. Dündar, Statistical convergence of double sequences of functions and some properties in 2-normed spaces, Facta Univ. Ser. Math. Inform., 33(5) (2018), 705–719.
• [13] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5 (1953), 29–49.
• [14] W.H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Can. J. Math., 25 (1973), 973–978.
• [15] I.J. Maddox, Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc., 100(1) (1986), 161–166.
• [16] I.J. Maddox, Inclusions between FK spaces and Kuttner’s theorem, Math. Proc. Cambridge Philos. Soc., 101(3) (1987), 523–527.
• [17] A. Aizpuru, M.C. Listan-Garcia, F. Rambla-Barreno, Density by moduli and statistical convergence, Quaest. Math., 37(4) (2014), 525–530.
• [18] A. Aizpuru, M.C. Listan-Garcia, F. Rambla-Barreno, Double density by moduli and statistical convergence, Bull. Belg. Math. Soc. Simon Stevin, 19(4) (2012), 663–673.
• [19] V.K. Bhardwaj, S. Dhawan, f-statistical convergence of order a and strong Ces`aro summability of order a with respect to a modulus, J. Ineq. Appl., 2015(332) (2015).
• [20] S. Hilger, Analysis on measure chains-a unified approach to continuous and discrete calculus, Results Math., 18(1-2) (1990), 18–56.
• [21] M. Bohner, A. Peterson, Dynamic Equations On Time Scales: An Introduction With Applications, Birkh¨auser, Boston, 2001.
• [22] R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl., 4(4) (2001), 535–557.
• [23] G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285(1) (2003), 107–127.
• [24] M. Bohner, G.S. Guseinov, Partial differentiation on time scales, Dynam. Syst. Appl., 13 (2004) , 351–379.
• [25] M. Bohner, G. S. Guseinov, Multiple Lebesgue integration on time scales, Adv. Difference Equ., 2006 (2006), Article ID 26391.
• [26] A. Cabada, D.R. Vivero, Expression of the Lebesgue D􀀀integral on time scales as a usual Lebesgue integral: Application to the calculus of D􀀀antiderivatives, Math. Comput. Model., 43(1-2) (2006), 194–207.
• [27] M.S. Seyyidoğlu, N.O. Tan, A note on statistical convergence on time scale, J. Inequal. Appl., 2012(219) (2012).
• [28] C. Turan, O. Duman, Statistical convergence on time scales and its characterizations, Springer Proc. Math. Stat., 41 (2013), 57–71.
• [29] C. Turan, O. Duman, Convergence methods on time scales, AIP Conf. Proc., 1558 (2013), 1120–1123.
• [30] C. Turan, O. Duman, Fundamental properties of statistical convergence and lacunary statistical convergence on time scales, Filomat, 31(14) (2017), 4455–4467.
• [31] Y. Altın, H. Koyunbakan, E. Yılmaz, Uniform statistical convergence on time scales, J. Appl. Math., 2014 (2014).
• [32] M. Çınar, E. Yılmaz, Y. Altın, T. Gülsen, Statistical convergence of double sequences on product time scales, Analysis, 39(3) (2019), 71–77.
• [33] B. Sözbir, S. Altundağ, Weighted statistical convergence on time scale, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 26 (2019), 137–143.
• [34] B. Sözbir and S. Altundağ, ab-statistical convergence on time scales, Facta Univ. Ser. Math. Inform., 35(1) (2020), 141–150.
• [35] B. Sözbir, S. Altundağ, M. Başarır, On the (Delta,f)-lacunary statistical convergence of the functions, Maltepe J. Math., 2(1) (2020), 1–8.
• [36] N. Turan, M. Başarır, On the ${\Delta _g}$-statistical convergence of the function defined time scale, AIP Conf. Proc., 2183, 040017 (2019), https://doi.org/10.1063/1.5136137.
• [37] N. Tok, M. Başarır, On the $\lambda _h^\alpha$-statistical convergence of the functions defined on the time scale, Proc. Int. Math. Sci., 1(1) (2019), 1–10.
• [38] M. Başarır, A note on the $\left( {\theta ,\varphi } \right)$-statistical convergence of the product time scale, Konuralp J. Math., 8(1) (2020), 192–196.
• [39] M. Başarır, A note on the $\left( {\lambda ;v} \right)_h^\alpha $-statistical convergence of the functions defined on the product of time scales, Azerbaijan Journal of Mathematics, 2020, under communication.