Asymptotically I-Cesaro equivalence of sequences of sets

Year 2018, Volume: 1 Issue: 2, 101 - 105, 26.06.2018

Erdinç Dundar , Uğur Ulusu

https://doi.org/10.32323/ujma.409463

Cited By: 6

Abstract

In this paper, we defined concepts of asymptotically $\mathcal{I}$-Cesaro equivalence and investigate the relationships between the concepts of asymptotically strongly $\mathcal{I}$-Cesaro equivalence, asymptotically strongly $\mathcal{I}$-lacunary equivalence, asymptotically $p$-strongly $ \mathcal{I}$-Cesaro equivalence and asymptotically $\mathcal{I}$-statistical equivalence of sequences of sets.

Keywords

Asymptotically equivalence, Ces\`{a}ro summability, statistical convergence, lacunary sequence, ideal convergence, sequences of sets, Wijsman convergence.

References

• [1] M. Baronti and P. Papini, Convergence of sequences of sets, In: Methods of Functional Analysis in Approximation Theory (pp. 133-155), ISNM 76, Birkhauser, Basel (1986).
• [2] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Aust. Math. Soc., 31 (1985), 421–432.
• [3] G. Beer, Wijsman convergence: A survey, Set-Valued Anal., 2 (1994), 77–94.
• [4] P. Das, E. Savas¸ and S. Kr. Ghosal, On generalizations of certain summability methods using ideals, Appl. Math. Letters, 24(9) (2011), 1509–1514.
• [5] H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244.
• [6] J. A. Fridy and C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160(1) (1993), 43–51.
• [7] Ö . Kişi and F. Nuray, New convergence definitions for sequences of sets, Abstr. Appl. Anal., 2013 (2013), Article ID 852796, 6 pages. http://dx.doi.org/10.1155/2013/852796.
• [8] Ö . Kişi, E. Savas¸ and F. Nuray, On asymptotically I-lacunary statistical equivalence of sequences of sets, (submitted for publication).
• [9] P. Kostyrko, T. ˇSalat and W. Wilczy´nski, I-Convergence, Real Anal. Exchange, 26(2) (2000), 669–686.
• [10] M. Marouf, Asymptotic equivalence and summability, Int. J. Math. Sci., 16(4) (1993), 755-762.
• [11] F. Nuray and B. E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math., 49 (2012), 87–99.
• [12] R. F. Patterson, On asymptotically statistically equivalent sequences, Demostratio Mathematica 36(1) (2003), 149-153.
• [13] R. F. Patterson and E. Savas¸, On asymptotically lacunary statistically equivalent sequences, Thai J. Math., 4(2) (2006), 267–272.
• [14] E. Savas¸, On I-asymptotically lacunary statistical equivalent sequences, Adv. Differ. Equ., 111 (2013), 7 pages. doi:10.1186/1687-1847-2013-111.
• [15] E. Savas¸ and P. Das, A generalized statistical convergence via ideals, Appl. Math. Letters, 24(6) (2011), 826–830.
• [16] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361–375.
• [17] U. Ulusu and E. Dündar, I-lacunary statistical convergence of sequences of sets, Filomat, 28(8) (2014), 1567–1574. DOI 10.2298/FIL1408567U.
• [18] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequence of sets, Progress in Applied Mathematics, 4(2) (2012), 99–109.
• [19] U. Ulusu and F. Nuray, On asymptotically lacunary statistical equivalent set sequences, Journal of Mathematics, 2013 (2013), Article ID 310438, 5 pages. http://dx.doi.org/10.1155/2013/310438.
• [20] U. Ulusu and Ö . Kişi, I-Cesa`ro summability of sequences of sets, Electronic Journal of Mathematical Analysis and Applications, 5(1) 2017, 278–286.
• [21] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc., 70(1) (1964), 186–188.
• [22] R. A. Wijsman, Convergence of Sequences of Convex Sets, Cones and Functions II, Trans. Amer. Math. Soc., 123(1) (1966), 32–45.