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Statistical convergence of sequences of sets in hyperspaces

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

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.

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

Abstract

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.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Sevda Sağıroğlu

Mehmet Ünver

Publication Date August 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 4

Cite

APA Sağıroğlu, S., & Ünver, M. (2018). Statistical convergence of sequences of sets in hyperspaces. Hacettepe Journal of Mathematics and Statistics, 47(4), 889-896.
AMA Sağıroğlu S, Ünver M. Statistical convergence of sequences of sets in hyperspaces. Hacettepe Journal of Mathematics and Statistics. August 2018;47(4):889-896.
Chicago Sağıroğlu, Sevda, and Mehmet Ünver. “Statistical Convergence of Sequences of Sets in Hyperspaces”. Hacettepe Journal of Mathematics and Statistics 47, no. 4 (August 2018): 889-96.
EndNote Sağıroğlu S, Ünver M (August 1, 2018) Statistical convergence of sequences of sets in hyperspaces. Hacettepe Journal of Mathematics and Statistics 47 4 889–896.
IEEE S. Sağıroğlu and M. Ünver, “Statistical convergence of sequences of sets in hyperspaces”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, pp. 889–896, 2018.
ISNAD Sağıroğlu, Sevda - Ünver, Mehmet. “Statistical Convergence of Sequences of Sets in Hyperspaces”. Hacettepe Journal of Mathematics and Statistics 47/4 (August 2018), 889-896.
JAMA Sağıroğlu S, Ünver M. Statistical convergence of sequences of sets in hyperspaces. Hacettepe Journal of Mathematics and Statistics. 2018;47:889–896.
MLA Sağıroğlu, Sevda and Mehmet Ünver. “Statistical Convergence of Sequences of Sets in Hyperspaces”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 4, 2018, pp. 889-96.
Vancouver Sağıroğlu S, Ünver M. Statistical convergence of sequences of sets in hyperspaces. Hacettepe Journal of Mathematics and Statistics. 2018;47(4):889-96.