Deferred Statistical Convergence in Metric Spaces

Abstract

In this paper, the concept of deferred statistical convergence is generalized to general metric spaces, and some inclusion relations between deferred strong Ces\`{a}ro summability and deferred statistical convergence are given in general metric spaces.

Keywords

• Metric Space,Statistical Convergence,Deferred statistical convergence

References

1. [1] A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, London and New York, 1979.
2. [2] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
3. [3] H. Fast, Sur la convergence statistique, Colloq. Math.,2 (1951), 241–244.
4. [4] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
5. [5] S. Gupta, V. K. Bhardwaj, On deferred f-statistical convergence, Kyungpook Math. J. 58(1) (2018), 91–103.
6. [6] N. L. Braha, H. M. Srivastava, S. A. Mohiuddine, A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean, Appl. Math. Comput., 228 (2014), 162–169.
7. [7] M. Çınar, M. Karaka¸s, M. Et, On pointwise and uniform statistical convergence of order $\alpha$ for sequences of functions, Fixed Point Theory Appl. 33(2013), 11.
8. [8] J. S. Connor, The Statistical and strong p-Cesàro convergence of sequences, Analysis, 8 (1988), 47–63.

9. [9] M. Et, A. Alotaibi, S. A. Mohiuddine, On $(\Delta^{m},I)-$-statistical convergence of order $\alpha$; The Scientific World Journal, 2014, 535419 DOI: 10.1155/2014/535419.
10. [10] M. Et, S. A. Mohiuddine, A. Alotaibi, On $\lambda $-statistical convergence and strongly $\lambda -$summable functions of order $\alpha$, J. Inequal. Appl. 469 (2013), 8.
11. [11] M. Et, B. C. Tripathy, A. J. Dutta, On pointwise statistical convergence of order $\alpha$ of sequences of fuzzy mappings, Kuwait J. Sci. 41(3) (2014), 17–30.
12. [12] M. Et, R. Colak, Y. Altın, Strongly almost summable sequences of order $\alpha$; Kuwait J. Sci. 41(2), (2014), 35–47.
13. [13] E. Savaş, M. Et, On $(\Delta_{\lambda}^{m},I)-$ statistical convergence of order $\alpha$, Period. Math. Hungar. 71(2) (2015), 135–145.
14. [14] J. A. Fridy, On statistical convergence, Analysis, 5 (1985), 301–313.
15. [15] M. I¸sık, K. E. Akba¸s, On Lamda-statistical convergence of order $\alpha$ in probability, J. Inequal. Spec. Funct. 8(4) (2017), 57–64.
16. [16] M. I¸sık, K. E. Et, On lacunary statistical convergence of order $\alpha$ in probability, AIP Conference Proceedings 1676, 020045 (2015); doi: http://dx.doi.org/10.1063/1.4930471.
17. [17] M. I¸sık, K. E. Akbaş, On Asymptotically Lacunary Statistical Equivalent Sequences of Order $\alpha$ in Probability, ITM Web of Conferences 13, 01024 (2017). DOI: 10.1051/itmconf/20171301024.
18. [18] S. A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Anal., 2002 (2012), Article ID 719729, 9 pp.
19. [19] M. Mursaleen, A. Khan, H. M. Srivastava, K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput., 219 (2013), 6911–6918.
20. [20] F. Nuray, Lamda-strongly summable and $\lambda-$-statistically convergent functions, Iran. J. Sci. Technol. Trans. A Sci., 34 (2010), 335–338.
21. [21] F. Nuray, B. Aydin, Strongly summable and statistically convergent functions, Inform. Technol. Valdymas 1(30) (2004), 74–76.
22. [22] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150.
23. [23] H. Şengül, M. Et, On I-lacunary statistical convergence of order $\alpha$ of sequences of sets, Filomat 31(8) (2017), 2403–2412.
24. [24] H. Şengül, On Wijsman I-lacunary statistical equivalence of order $(\eta,\mu)$, J. Inequal. Spec. Funct. 9(2) (2018), 92–101.
25. [25] H. Şengül, On $S_{\alpha}^{\beta}\left( \theta\right) -$ convergence and strong $N_{\alpha}^{\beta}\left( \theta,p\right) -$ summability, J. Nonlinear Sci. Appl. 10(9) (2017), 5108–5115.
26. [26] H. Şengül, M. Et, Lacunary statistical convergence of order $(\alpha,\beta)$ in topological groups, Creat. Math. Inform. 2683 (2017), 339–344.
27. [27] H. M. Srivastava, M. Mursaleen, A. Khan, Generalized equi-statistical convergence of positive linear operators and associated approximation theorems, Math. Comput. Modelling 55 (2012), 2040–2051.
28. [28] H. M. Srivastava, M. Et, Lacunary statistical convergence and strongly lacunary summable functions of order $\alpha$; Filomat 31(6) (2017), 1573–1582.
29. [29] R. P. Agnew, On deferred Cesàro mean, Ann. Math.,33 (1932), 413-421.
30. [30] M. Küçükaslan, M. Yılmaztürk On deferred statistical convergence of sequences, Kyungpook Math. J. 56 (2016), 357-366.