ON λ-ASYMPTOTICALLY WIJSMAN GENERALIZED STATISTICAL CONVERGENCE OF SEQUENCES OF SETS

Hazarika, Bipan; Esi, Ayhan

doi:10.2478/tmmp-2013-0025

Abstract

The concept of Wijsman statistical convergence was defined by [Nuray, F.-Rhoades, B. E.: Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 1-9]. In this paper we present three definitions which are a natural combination of the definition of asymptotic equivalence, statistical convergence, generalized statistical convergence and Wijsman convergence. In addition, we also present asymptotically equivalent sequences of sets in sense of Wijsman and study some properties of this concept.

References

[1] AUBIN, J. P.-FRANKOWSKA, H.: Set-valued Analysis, Birkhauser, Boston, 1990.
Search in Google Scholar
[2] BARONTI, M.-PAPINI, P.: Convergence of sequences of sets, in: Methods of Functional Analysis in Approximation Theory, Proc. Int. Conf., Bombay, 1985, Birkhauser, Basel, 1986, pp. 133-155.
Search in Google Scholar
[3] BEER, G.: Convergence of continuous linear functionals and their level sets, Archiv der Mathematik 52 (1989), 482-491.10.1007/BF01198356
Search in Google Scholar
[4] BEER, G.: On convergence of closed sets in a metric space and distance functions, Bull.Austral. Math. Soc. 31 (1985), 421-432.10.1017/S0004972700009370
Search in Google Scholar
[5] BORWEIN, J. M.-VANDERWERFF, J. D.: Dual Kadec-Klee norms and the relationship between Wijsman, slice and Mosco convergence, Mich. Math. J. 41 (1994), 371-387.10.1307/mmj/1029005003
Search in Google Scholar
[6] BUCK, R.C.: Generalized asymptotic density, Amer. J. Math. 75 (1953), 335-346.10.2307/2372456
Search in Google Scholar
[7] ESI, A.-HAZARIKA, B.: λ-statistical convergence of sequences of sets (preprint).
Search in Google Scholar
[8] FAST, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 241-244.10.4064/cm-2-3-4-241-244
Search in Google Scholar
[9] FRIDY, J. A.: On statistical convergence, Analysis 5 (1985), 301-313.10.1524/anly.1985.5.4.301
Search in Google Scholar
[10] HAZARIKA, B.-ESI, A.: Statistically almost λ-convergence of sequences of sets, European J. of Pure Appl. Math. 6 (2013), 137-146.
Search in Google Scholar
[11] LEINDLER, L.: ¨Uber die de la Vall´ee-Pousinsche Summeierbarkeit allgemeiner Orthogonalreihen, Acta Math. Acad. Sci. Hungar. 16 (1965), 375-387.10.1007/BF01904844
Search in Google Scholar
[12] MAROUF, M. S.: Asymptotic equivalence and summability, Internat. J. Math.Math. Sci. 16 (1993), 755-762.10.1155/S0161171293000948
Search in Google Scholar
[13] MURSALEEN, M.: λ-statistical convergence, Math. Slovaca 50 (2000), 111-115.
Search in Google Scholar
[14] NURAY, F.-RHOADES, B. E.: Statistical convergence of sequences of sets, Fasc. Math. 49 (2012), 1-9.
Search in Google Scholar
[15] PATTERSON, R. F.: On asymptotically statistically equivalent sequences, Demonstratio Math. 36 (2003), 149-153.
Search in Google Scholar
[16] POBYVANCTS, I. P.: Asymptotic equivalence of some linear transformation defined by a nonnegative matrix and reduced to generalized equivalence in the sense of Ces`aro and Abel, Mat. Fiz. 28 (1980), 83-87.
Search in Google Scholar
[17] ŠALÀT, T.: On statistical convergence of real numbers, Math. Slovaca 30 (1980), 139-150.
Search in Google Scholar
[18] SCHOENBERG, I. J.: The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361-375.10.1080/00029890.1959.11989303
Search in Google Scholar
[19] STEINHAUS, H.: Sur la convergence ordinate et la convergence asymptotique, Colloq.Math. 2 (1951), 73-84.10.4064/cm-2-2-98-108
Search in Google Scholar
[20] ULUSU, U.-NURAY, F.: Lacunary statistical convergence of sequence of sets, Progr.Appl. Math. 4 (2012), 99-109.
Search in Google Scholar
[21] ULUSU, U.-NURAY, F.: On asymptotically lacunary statistical equivalent set sequences, J. Math. 2013 (2013), 5 p.10.1155/2013/310438
Search in Google Scholar
[22] WIJSMAN, R. A.: Convergence of sequences of convex sets, cones and functions, Bull.Amer. Math. Soc. 70 (1964), 186-188.10.1090/S0002-9904-1964-11072-7
Search in Google Scholar
[23] WIJSMAN, R. A.: Convergence of sequences of convex sets, cones and functions II, Trans.Amer. Math. Soc. 123 (1966), 32-45.10.1090/S0002-9947-1966-0196599-8
Search in Google Scholar
[24] ZYGMUND, A.: Trigonometric Series. Cambridge Univ. Press, Cambridge, UK, 979.
Search in Google Scholar

ON λ-ASYMPTOTICALLY WIJSMAN GENERALIZED STATISTICAL CONVERGENCE OF SEQUENCES OF SETS

Abstract

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