On $\rho -$ Statistical convergence of sequences of Sets

Abstract

In this paper we introduce the concepts of Wijsman $\rho-$statistical convergence, Wijsman strongly $\rho-$statistical convergence and Wijsman $\rho-$strongly $p-$ summability. Also, the relationship between these concepts are given. \newline\newline \textbf{Keywords:} Ces\`{a}ro summability, Statistical convergence, Strongly $p-$Ces\`{a}ro summability, Wijsman convergence.

Keywords

• Cesaro summability
• Statistical convergence
• Strongly p-Cesaro summability
• Wijsman convergence

References

1. 1 H. Altınok, M. Et, R. Çolak, Some remarks on generalized sequence space of bounded variation of sequences of fuzzy numbers, Iran. J. Fuzzy Syst. 11(5) (2014), 39–46.
2. 2 V. K. Bhardwaj, S. Dhawan, f-statistical convergence of order $\alpha $ and strong Cesàro summability of order $\alpha $ with respect to a modulus, J. Inequal. Appl. 332 (2015), 14 pp.
3. 3 A. Caserta, G. Di Maio, L. D. R. Kocinac, Statistical convergence in function spaces, Abstr. Appl. Anal., (2011), Article ID 420419, 11 pp.
4. 4 J. S. Connor, The Statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), 47–63.
5. 5 H. Çakallı, Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26(2) (1995), 113–119.
6. 6 H. Çakallı , B. Hazarika, Ideal quasi-Cauchy sequences, J. Inequal. Appl. 234 (2012), 11 pp.
7. 7 H. Çakallı, A variation on ward continuity, Filomat 27(8) (2013), 1545–1549.
8. 8 H. Çakallı, A variation on statistical ward continuity, Bull. Malays. Math. Sci. Soc. 40 (2017), 1701-1710.

9. 9 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 pp.
10. 10 R. Çolak, Statistical convergence of order $\alpha $, Modern Methods in Analysis and Its Applications, New Delhi, India: Anamaya Pub. 2010 (2010) 121–129.
11. 11 R. Çolak, On $\lambda $-Statisitical Convergence, Conference on Summability and Applications, (2011) Istanbul Commerce Univ. May 12-13, ˙Istanbul.
12. 12 M. Et, A. Alotaibi, S. A. Mohiuddine, On $(\Delta ^{m},I)-$ statistical convergence of order $\alpha$, The Scientific World Journal, (2014), Article ID 535419, 5 pages.
13. 13 M. Et, H. ¸ Sengül, Some Cesaro-type summability spaces of order $\alpha$ and lacunary statistical convergence of order $\alpha$, Filomat, 28(8), (2014), 1593–1602.
14. 14 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.
15. 15 M. Et, R. Çolak, Y. Altın, Strongly almost summable sequences of order $\alpha$, Kuwait J. Sci. 41(2) (2014), 35–47.
16. 16 H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
17. 17 J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313.
18. 18 A. D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129–138.
19. 19 M. Işık, K. E. Akbaş, On $\lambda -$statistical convergence of order $\alpha $ in probability, J. Inequal. Spec. Funct. 8(4) (2017), 57–64.
20. 20 F. Nuray, B. E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 87–99.
21. 21 T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), 139–150.
22. 22 E. Savaş, M. Et, On $(\Delta _{\lambda }^{m},I)-$\statistical convergence of order $\alpha $, Period. Math. Hungar. 71(2) (2015), 135–145.
23. 23 I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375.
24. 24 H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951), 73–74.
25. 25 H. Şengül, Some Cesàro-type summability spaces defined by a modulus function of order $\left( \alpha ,\beta \right) $, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 66(2) (2017), 80–90.
26. 26 U. Ulusu, F. Nuray, Lacunary statistical convergence of sequence of sets, Prog. Appl. Math. 4(2) (2012), 99–109.
27. 27 U. Ulusu, E. Dündar, I-lacunary statistical convergence of sequences of sets, Filomat 28(8) (2014), 1567–1574.