On statistical convergence in quasi-metric spaces. (English) Zbl 1426.54023
Summary: A quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.
MSC:
| 54E35 | Metric spaces, metrizability |
| 54A20 | Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) |
Keywords:
quasi-metric space; asymptotic density; statistical convergence; forward Cauchy sequences; backward Cauchy sequencesReferences:
| [1] | Wilson W. A., On quasi-metric spaces, Amer. J. Math., 1931, 53, 675-684 · JFM 57.0735.02 |
| [2] | Reilly I. L., Vamanamurthy M. K., On oriented metric spaces, Math. Slovaca, 1984, 34(3), 299-305 · Zbl 0599.54031 |
| [3] | Kelly J. C., Bitopological spaces, J. Proc. London Math. Soc., 1963, 13, 71-89 · Zbl 0107.16401 |
| [4] | Reilly I. L., Subrahmanyam P. V., Vamanamurthy M. K., Cauchy sequences in quasi-pseudo-metric spaces, Monatsh. Math., 1982, 93(2), 127-140 · Zbl 0472.54018 |
| [5] | Alegre C., Ferrando I., Garcia-Rafl L. M., Sánchez Pérez E. A., Compactness in asymmetric normed spaces, Topology Appl., 2008, 155(6), 527-539 · Zbl 1142.46004 |
| [6] | Cobzaş Ş., Functional Analysis in Asymmetric Normed Spaces, Birkhäuser, Basel, 2013 · Zbl 1266.46001 |
| [7] | Collins J., Zimmer J., An asymmetric Arzelà-Ascoli theorem, Topology Appl., 2007, 154(11), 2312-2322 · Zbl 1123.26002 |
| [8] | García-Rafl L. M., Compactness and finite dimension in asymmetric normed linear spaces, Topology Appl., 2005, 153, 844-853 · Zbl 1101.46017 |
| [9] | Künzi H. P., Complete quasi-pseudo-metric spaces, Acta Math. Hungar., 1992, 59(1-2), 121-146 · Zbl 0784.54026 |
| [10] | Romaguera S., Left K-completeness in quasi-metric spaces, Math. Nachr., 1992, 157, 15-23 · Zbl 0784.54027 |
| [11] | Romaguera S., Gutiérrez A., A note on Cauchy sequences in quasi pseudo metric spaces, Glas. Mat. Ser. III, 1986, 21(41)(1), 191-200 · Zbl 0606.54025 |
| [12] | Zygmund A., Trigonometric Series, 3rd ed., Cambridge University Press, Cambridge, 2002 · Zbl 1084.42003 |
| [13] | Fast H., Sur la convergence statistique, Colloq. Math., 1951, 2, 241-244 · Zbl 0044.33605 |
| [14] | Schoenberg I. J., The integrability of certain functions and related summability methods, Amer. Math. Monthly, 1959, 66, 361-375 · Zbl 0089.04002 |
| [15] | Fridy J. A., On statistical convergence, Analysis, 1985, 5, 301-313 · Zbl 0588.40001 |
| [16] | Fridy J. A., Orhan C., Lacunary statistical summability, J. Math. Anal. Appl., 1993, 173, 497-504 · Zbl 0786.40004 |
| [17] | Kirişci M., Karaisa A., Fibonacci statistical convergence and Korovkin type approximation theorems, J. Inequal. Appl., 2017, 2017(1), 229 · Zbl 1425.40003 |
| [18] | Küçükaslan M., Değer U., Dovgoshey O., On the statistical convergence of metric-valued sequences, Ukrainian Math. J., 2014, 66(5), 796-805 · Zbl 1325.40008 |
| [19] | Di Maio G., Kočinac L. D. R., Statistical convergence in topology, Topology Appl., 2008, 156, 28-45 · Zbl 1155.54004 |
| [20] | Mursaleen M., Edely O. H. H., Generalized statistical convergence, Inform. Sci., 2004, 162, 287-294 · Zbl 1062.40003 |
| [21] | Dündar E., Ulusu U., Asymptotically I-Cesàro equivalence of sequences of sets, Univers. J. Math. Appl., 2018, 1, 101-105 |
| [22] | Kişi Ö, Güler E., 𝒥-Cesàro summability of a sequence of order α of random variables in probability, Fund. Jo. Mathe. Appl., 2018, 1, 157-161 |
| [23] | Das P., Som S., Ghosal S., Karakaya V., A notion of αβ-statistical convergence of order γ in probability, Kragujevac J. Math., 2018, 42(1), 51-67 · Zbl 1488.40019 |
| [24] | Edely O. H. H., Mohiuddine S. A., Noman A. K., Korovkin type approximation theorems obtained through generalized statistical convergence, Appl. Math. Lett., 2010, 23, 1382-1387 · Zbl 1206.40003 |
| [25] | Belen C., Mohiuddine S. A., Generalized weighted statistical convergence and application, Appl.Math. Comput., 2013, 219, 9821-9826 · Zbl 1308.40003 |
| [26] | Kadak U., Mohiuddine S. A., Generalized statistically almost convergence based on the difference operator which includes the (p, q)-Gamma function and related approximation theorems, Results Math., 2018, 73(9), 1-31 · Zbl 1390.41008 |
| [27] | Toyganözü Z. H., Pehlivan S., Some results on exhaustiveness in asymmetric metric spaces, Filomat, 2015, 29(1), 183-192 · Zbl 1453.54002 |
| [28] | Das P., Ghosal S., Pal S. K., Extending asymmetric convergence and Cauchy condition using ideals, Math. Slovaca, 2013, 63(3), 545-562 · Zbl 1349.40021 |
| [29] | Li K., Lin S., Ge Y., On statistical convergence in cone metric spaces, Topology Appl., 2015, 196, 641-651 · Zbl 1343.40003 |
| [30] | Fridy J. A., Statistical limit points, Proc. Amer. Math. Soc., 1993, 118(4), 1187-1192 · Zbl 0776.40001 |
| [31] | Burgin M., Duman O., Statistical convergence and convergence in statistics, arXiv:math/0612179 [math.GM] · Zbl 1175.26064 |
| [32] | Šalát T., On statistically convergent sequences of real numbers, Math. Slovaca, 1980, 30(2), 139-150 · Zbl 0437.40003 |
| [33] | Künzi H. P., A note on sequentially compact quasi-pseudo-metric spaces, Monatsh. Math., 1983, 95, 219-220 · Zbl 0505.54026 |
| [34] | Künzi H. P., Mršević M., Reilly I. L., Vamanamurthy M. K., Convergence, precompactness and symmetry in quasi-uniform spaces, Math. Japon., 1993, 38, 239-253 · Zbl 0783.54022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
