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

Bayram Sözbir; Selma Altundağ; Metin Basarır

doi:10.32323/ujma.743949

Research Article

Year 2020, , 138 - 143, 23.12.2020

Bayram Sözbir , Selma Altundağ , Metin Basarır

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

Cited By: 1

Abstract

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.

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

Year 2020, , 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.

There are 39 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Articles
Authors	Bayram Sözbir 0000-0002-9475-7180 Selma Altundağ 0000-0002-5893-9868 Metin Basarır 0000-0002-4341-4399
Publication Date	December 23, 2020
Submission Date	May 28, 2020
Acceptance Date	October 22, 2020
Published in Issue	Year 2020

Cite

APA	Sözbir, B., Altundağ, S., & Basarır, M. (2020). On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale. Universal Journal of Mathematics and Applications, 3(4), 138-143. https://doi.org/10.32323/ujma.743949
AMA	Sözbir B, Altundağ S, Basarır M. On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale. Univ. J. Math. Appl. December 2020;3(4):138-143. doi:10.32323/ujma.743949
Chicago	Sözbir, Bayram, Selma Altundağ, and Metin Basarır. “On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale”. Universal Journal of Mathematics and Applications 3, no. 4 (December 2020): 138-43. https://doi.org/10.32323/ujma.743949.
EndNote	Sözbir B, Altundağ S, Basarır M (December 1, 2020) On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale. Universal Journal of Mathematics and Applications 3 4 138–143.
IEEE	B. Sözbir, S. Altundağ, and M. Basarır, “On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale”, Univ. J. Math. Appl., vol. 3, no. 4, pp. 138–143, 2020, doi: 10.32323/ujma.743949.
ISNAD	Sözbir, Bayram et al. “On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale”. Universal Journal of Mathematics and Applications 3/4 (December 2020), 138-143. https://doi.org/10.32323/ujma.743949.
JAMA	Sözbir B, Altundağ S, Basarır M. On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale. Univ. J. Math. Appl. 2020;3:138–143.
MLA	Sözbir, Bayram et al. “On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale”. Universal Journal of Mathematics and Applications, vol. 3, no. 4, 2020, pp. 138-43, doi:10.32323/ujma.743949.
Vancouver	Sözbir B, Altundağ S, Basarır M. On the $\Delta _{{\Lambda ^2}}^f$-Statistical Convergence on Product Time Scale. Univ. J. Math. Appl. 2020;3(4):138-43.