Deferred Statistical $r$-Convergence of Sequences of Sets

Year 2025, Volume: 13 Issue: 2, 125 - 133, 31.10.2025

Naveen Sharma , Sandeep Kumar

Abstract

In the present paper, we introduce and study the concept of Wijsman deferred statistical $r$-convergence of sequences of sets and have its characterization in terms of deferred statistically dense subsequences. Beside this, we explore the concept of strongly deferred Cesàro summability and its relation with the newly introduced notion of Wijsman deferred statistical $r$-convergence.

Keywords

Statistical convergence , Deferred statistical convergence , Wijsman convergence , Wijsman statistical convergence

References

• [1] R. P. Agnew, On deferred Ces`aro mean, Ann. Math., 33(1932), 413-421.
• [2] M. Altınok, B. Inan and M. Kucukaslan, On asymptotically Wijsman deferred statistical equivalence of sequence of sets Thai J. Math., 18(2) (2020), 803-817.
• [3] S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optim., 29(3-4) (2008), 291-303.
• [4] E. Bayram, A. Aydin and M. Kucukaslan, Weighted statistical rough convergence in normed spaces, Maejo Int. J. Sci. Technol., 18(2) (2024), 178-192.
• [5] V. K. Bhardwaj and S. Dhawan, Density by moduli and Wijsman lacunary statistical convergence of sequences of sets, J. Ineq. Appl., 2017, 1-20.
• [6] R. C. Buck, Generalized asymptotic density, Amer. J. Math., 75(2) (1953), 335-346.
• [7] A. Esi, N. L. Braha and A. Rushiti, Wijsman l-statistical convergence of interval numbers, Bol. Soc. Parana. Mat., 35 (2017), 9-18.
• [8] M. Et and M. C. Yilmazer, On deferred statistical convergence of sequences of sets, AIMS Mathematics, 5(3) (2020), 2143-2152.
• [9] H. Fast, Sur la convergence statistique, Colloq. Math., 2(3-4) (1951), 241-244.
• [10] A. R. Freedman, J. J. Sember and M. Raphael, Some Ces`aro-type summability spaces, Proc. Londan Math. Soc., 37(3) (1978), 508-520.
• [11] J. A. Fridy, On statistical convergence, Analysis, 5(4) (1985), 301-314.
• [12] J. A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173(2) (1993), 497-504.
• [13] E. Gülle and U. Ulusu, Wijsman deferred invariant statistical and strong $p$-deferred invariant equivalence of order $\alpha$, Fundam. J. Math. Appl., 6(4) (2023), 211-217.
• [14] S. Gupta and V. K. Bhardwaj, On deferred f-statistical convergence, Kyungpook Math. J., 58 (2018), 91-103.
• [15] B. Hazarika and A. Esi, On l-asymptotically Wijsman generalized statistical convergence of sequences of sets, Tatra Mt. Math. Publ.-Number Theory, 56 (2013), 67-77.
• [16] B. Hazarika, A. Esi and N. L. Braha, On asymptotically Wijsman s-statistical convergence of set sequences, J. Math. Anal., 4(3) (2013), 33-46.
• [17] M. Küçükaslan and M. Yılmaztürk, On deferred statistical convergence of sequences, Kyungpook Math. J., 56 (2016) 357-366.
• [18] M. Mursaleen, l-statistical convergence, Math. Slovaca, 50 (2000), 111-115.
• [19] F. Nuray and B. E. Rhoades, Statistical convergence of sequence of sets, Fasc. Math., 49 (2012), 87-99.
• [20] H. X. Phu, Rough convergence in normed linear spaces, Numer. Funct. Anal. Optim., 22(1-2) (2001), 199-222.
• [21] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca, 30(2) (1980), 139-150.
• [22] E. Savas¸, On I-lacunary statistical convergence of order a for sequences of sets, Filomat, 29(6) (2015), 1223-1229.
• [23] I. J. Schoenberg, The integrability of certain functions and related summability methods, The Amer. Math. monthly, 66(5) (1959), 361-375.
• [24] H. Steinhaus, Sur la convergence ordinaireet la convergence asymptotique, Colloq. Math., 2(1) (1951), 73-74.
• [25] O. Talo, Y. Sever and F. Bas¸ar, On Statistically convergent sequences of closed sets, Filomat, 30(6) (2016), 1497–1509.
• [26] U. Ulusu and E. Dundar, I-lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567-1574.
• [27] U. Ulusu and F. Nuray, Lacunary statistical summability of sequences of sets, Konuralp J. Math., 3(2) (2015), 176-184.
• [28] U. Ulusu and E. Savas¸, An extension of asymptotically lacunary statistical equivalence set sequences, J. Ineq. Appl., 2014, 1-8.
• [29] R. A. Wijsman, Convergence of sequence of convex sets, cones and functions, Bull. Amer. Math. Soc., 70 (1964), 186-188.
• [30] R. A. Wijsman, Convergence of sequence of convex sets, cones and functions II, Trans. Amer. Math. Soc., 123(1) (1966), 32-45.
• [31] M. Yılmazt¨urk and M. Kucukaslan, On strongly deferred Cesaro summability and deferred statistical convergence of the sequences, Bitlis Eren Univ. J. Sci. Technol., 3(1) (2013) 22-25.
• [32] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, UK 1979.