Statistical convergence of sequences of sets in hyperspaces

Year 2018, Volume: 47 Issue: 4, 889 - 896, 01.08.2018

Sevda Sağıroğlu Mehmet Ünver

Abstract

The concept of statistical convergence in an arbitrary topological space is nothing new, it is actually a self-evident concept that comes through the structure of that space. In this paper, by considering the well known topologies on hyperspaces, we investigate the characterizations of statistical convergence of sequences of sets in the realm of these structures.

Keywords

Hyperspaces, weak topologies, statistical convergence

References

• Attouch, H., Lucchetti, R. and Wets, R. The topology of the $\rho$-Hausdorff distance, Ann. Mat. Pura. Appl. 160, 303320, 1991.
• Beer, G. Metric spaces with nice closed balls and distance function for closed sets, Bull. Austral. Math. Soc. 35, 8196, 1978.
• Beer, G. On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 95, 737739, 1988.
• Beer, G. An embedding theorem for the Fell topology, Michigan Math. J. 35, 39, 1988.
• Beer, G. Topologies on closed and convex sets, Math. App. 268, Kluwer Academic Publishers Group, Dordrecht, 1993.
• Cakalli, H. and Khan, M. K. Summability in topological spaces, Appl. Math. Lett. 24 (3), 348352, 2011.
• Cakalli, H. Lacunary statistical convergence in topological groups, Indian J. Pure Appl. Math. 26 (2), 113119, 1995.
• Connor, J. S. The statistical and strong p-Cesàro convergence of sequences, Analysis 8, 4763, 1988.
• Fast, H. Sur la convergence statistique, Colloq. Math. 2, 241244, 1951.
• Freedman, A.R. and Sember, J.J. Densities and summability, Pacic J. Math. 95, 293305, 1981.
• Fridy, J.A. On statistical convergence, Analysis 5, 301313, 1985.
• Fridy, J.A. and Miller H. I. A matrix characterization of statistical convergence, Analysis 11, 5966, 1991.
• Fridy, J.A. and Orhan, C. Statistical limit superior and limit inferior, Proc. Amer. Math. Soc. 125 (12), 36253631, 1997.
• Khan, M.K. and Orhan, C. Matrix characterization of A-statistical convergence, J. Math. Anal. Appl. 335, 406417, 2007.
• Kişi, Ö. and Nuray, F. New convergence definitions for sequences of sets, Abst. Appl. Anal. Volume 2013, Article ID 852796.
• Kolk, E. Matrix summability of statistically convergent sequences, Analysis 13 (1-2), 7783, 1993.
• Kostyrko, P., Macaj, M. and ’alát, T. I-convergence, Real Anal. Exchange 26 (2), 669686, 2000.
• Maddox, I.J. Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1), 141145, 1988.
• Maio, G. D. and Ko£inac, L.D.R. Statistical convergence in topology, Topology Appl. 156, 2845, 2008.
• Miller, H.I. A measure theoretical subsequence characterization of statistical convergence. Trans. Amer. Math. Soc. 347 (5), 18111819, 1995.
• Mosco, U. Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (4), 510585, 1969.
• Nuray, F. and Rhoades, B.E. Statistical convergence of sequences of sets, Fasc. Math. 49, 8799, 2012.
• Pancaroğlu, N. and Nuray, F. Invariant statistical convergence of sequences of sets with respect to a modulus function, Abst. Appl. Anal. Volume 2014, Article ID 818020.
• S’alát, T. On statistically convergent sequences of real numbers, Math. Slovaca, 30 (2), 139150, 1980.
• Savaş, E. On I-lacunary statistical convergence of order $\alpha$ for sequences of sets, Filomat, 29 (6), 2015.
• Unver, M., Khan, M. K. and Orhan, C. A-distributional summability in topological spaces, Positivity, 18 (1), 131145, 2014.
• Wets, R. Convergence of convex functions, variational inequalities, and convex optimization problems, Variational inequalities and complementarity problems 375403, 1980.
• Wijsman, R.A. Convergence of sequences of convex sets, cones and functions, II. Trans. Amer. Math. Soc. 123, 3245, 1966.