LACUNARY STATISTICAL SUMMABILITY OF SEQUENCES OF SETS

In this paper we de ne the WS?analog of the Cauchy criterion for convergence and show that it is equivalent to Wijsman lacunary statistical convergence. Also, Wijsman lacunary statistical convergence is compared to other summability methods which are de ned in this paper. After giving new de nitions for convergence, we prove a result comparing them. In addition, we give the relationship between Wijsman lacunary statistical convergence and Hausdorf lacunary statistical convergence.

Keywords:

Statistical convergence, lacunary sequence, sequence of sets, Wijsman convergence, Hausdor convergence Cesaro summability, almost convergence,

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[1] J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhauser, Boston, 1990.
[2] M. Baronti and P. Papini, Convergence of sequences of sets. In: Methods of Functional Analysis in Approximation Theory, ISNM 76, Birkhauser, Basel, 133-155, 1986.
[3] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Austral. Math. Soc., 31 (1985), 421-432.
[4] G. Beer, Wijsman convergence: A survey. Set-Valued Var. Anal., 2 (1994), 77-94.
[5] H. Fast, Sur la convergence statistique, Collog. Math., 2 (1951), 241-244.
[6] J.A. Fridy, On statistical convergence, Analysis, 5 (1985), 301-313.
[7] J.A. Fridy and C. Orhan, Lacunary statistical convergence, Paci c J. Math., 160(1) (1993), 43-51.
[8] J.A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173 (1993), 497-504.
[9] F. Nuray and B.E. Rhoades, Statistical convergence of sequences of sets, Fasc. Math., 49 (2012), 87-99.
[10] R.E. Powel and S.M. Shah, Summability theory and its applications, Van Nostrand- Rheinhold, London, 1972.
[11] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly, 66 (1959), 361-375.
[12] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Progress in Applied Mathematics, 4 (2012), 99-109.
[13] R.A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc., 70 (1964), 186-188.
[14] R. A. Wijsman, Convergence of sequences of Convex sets, Cones and Functions II, Trans. Amer. Math. Soc., 123 (1) (1966), 32-45.