\(A\)-statistically localized sequences in \(n\)-normed spaces. (English) Zbl 1489.40008
Summary: In [Izv. Vyssh. Uchebn. Zaved., Mat. 1974, No. 4(143), 45–54 (1974; Zbl 0292.54034)] L. N. Krivonosov defined the concept of localized sequence that is defined as a generalization of Cauchy sequence in metric spaces. In this present work, the \(A\)-statistically localized sequences in \(n\)-normed spaces are defined and some main properties of \(A\)-statistically localized sequences are given. Also, it is shown that a sequence is \(A\)-statistically Cauchy iff its \(A\)-statistical barrier is equal to zero. Moreover, we define the uniformly \(A\)-statistically localized sequences on \(n\)-normed spaces and investigate its relationship with \(A\)-statistically Cauchy sequences.
Keywords:
\(A\)-statistical convergence; \(n\)-normed spaces; \(A\)-statistical localor of the sequenceCitations:
Zbl 0292.54034References:
| [1] | Connor, B. J., The statistical and strong p-Cesaro convergence of sequences, Analysis (Munich), 8 (1988), 47-63. · Zbl 0653.40001 |
| [2] | Connor, J. , Kline, J., On statistical limit points and the consistency of statistical convergence, J. Math. Anal. Appl., 197 (1996), 392-399. · Zbl 0867.40001 |
| [3] | Fast, H., Sur la convergence statistique, Colloq. Math., 2 (1951), 241-244. · Zbl 0044.33605 |
| [4] | Fridy, J. A., On statistical convergence, Analysis (Munich), 5 (1985), 301-313. · Zbl 0588.40001 |
| [5] | Gähler, S., 2-metrische Räume und ihre topologische Struktur, Math. Nachr., 26 (1963), 115-148. · Zbl 0117.16003 |
| [6] | Gähler, S., Lineare 2-normierte räume, Math. Nachr., 28(1-2) (1964), 1-43. · Zbl 0142.39803 |
| [7] | Gähler, S., Untersuchungen über verallgemeinerte m-metrische räume. I., Math. Nachr., 40(1-3) (1969), 165-189 · Zbl 0182.56404 |
| [8] | Gähler, S., Siddiqi, A. H., Gupta, S.C., Contributions to non-archimedean functional analysis, Math. Nachr., 69 (1963), 162-171. · Zbl 0331.46012 |
| [9] | Gunawan, H., Mashadi, On n-normed Spaces, Int. J. Math. Sci., 27(10) (2001), 631-639. · Zbl 1006.46006 |
| [10] | Gürdal, M., Pehlivan, S., The statistical convergence in 2-Banach spaces, Thai. J. Math., 2(1) (2004), 107-113. · Zbl 1077.46005 |
| [11] | Gürdal, M., Pehlivan, S., The statistical convergence in 2-normed spaces, Southeast Asian Bull. Math., 33(2) (2009), 257-264. · Zbl 1212.40012 |
| [12] | Gürdal, M., Açık, I., On I-cauchy sequences in 2-normed spaces, Math. Inequal. Appl., 11(2) (2008), 349-354. · Zbl 1145.40001 |
| [13] | Gürdal, M., Şahiner, A., Statistical approximation with a sequence of 2-Banach spaces, Math. Comput. Modelling, 55(3-4) (2012), 471-479. · Zbl 1266.40009 |
| [14] | Gürdal, M., Şahiner, A., Açık, I., Approximation theory in 2-Banach spaces, Nonlinear Anal., 71(5-6) (2009), 1654-1661. · Zbl 1185.46004 |
| [15] | Gürdal, M., Yamancı, U., Statistical convergence and some questions of operator theory, Dynam. Syst. Appl., 24 (2015), 305-312. · Zbl 1343.47020 |
| [16] | 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. · Zbl 1390.41008 |
| [17] | Kolk, E., The statistical convergence in Banach spaces, Acta et Commentationes Universitatis Tartuensis, 928 (1991), 41-52. |
| [18] | Krivonosov, L. N., Localized sequences in metric spaces, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1974), 45-54; Soviet Math. (Iz. VUZ), 18(4), 37-44. · Zbl 0292.54034 |
| [19] | Mohiuddine, S. A., Statistical weighted A-summability with application to Korovkin’s type approximation theorem, J. Inequal. Appl., (2016), 2016:101. · Zbl 1333.41004 |
| [20] | Mohiuddine, S. A., Alamri, B. A. S., Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 113 (2019), 1955-1973. · Zbl 1435.40005 |
| [21] | Mohiuddine, S. A., Asiri, A., Hazarika, B., Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int. J. Gen. Syst., 48(5) (2019), 492-506. Mohiuddine, S. A., Hazarika, B., Alghamdi, M. A., Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems, Filomat, 33(14) (2019), 4549-4560. · Zbl 1499.40037 |
| [22] | Mohiuddine, S. A., Şevli, H., Cancan, M., Statistical convergence in fuzzy 2-normed space, J. Comput. Anal. Appl., 12(4) (2010), 787-798. Mursaleen, M., On statistical convergence in random 2-normed spaces, Acta Sci. Math. (Szeged), 76 (2010), 101-109. · Zbl 1198.40002 |
| [23] | Nabiev, A. A., Savaş, E., Gürdal, M., Statistically localized sequences in metric spaces. J. App. Anal. Comp., 9(2) (2019), 739-746. Nabiev, A. A., Savaş, E., Gürdal, M., I-localized sequences in metric spaces, Facta Univ. Ser. Math. Inform., 35(2) (2020), 459-469. · Zbl 1488.40027 |
| [24] | Rath, D., Tripathy, B.C., On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math., 25(4) (1994), 381-386. · Zbl 0812.46001 |
| [25] | Raymond, W. F., Cho, Ye. J., Geometry of linear 2-normed spaces, Huntington, N.Y. Nova Science Publishers, 2001. · Zbl 1051.46001 |
| [26] | Šalát, T., On statistically convergent sequences of real numbers, Mathematica Slovaca, 30(2) (1980), 139-150. · Zbl 0437.40003 |
| [27] | Savaş, E., A-statistical convergence of order α via ϕ-function, Appl. Anal. Discrete Math., 13 (2019), 918-926. · Zbl 1499.40044 |
| [28] | Savaş E. and Gürdal M., Certain summability methods in intuitionistic fuzzy normed spaces, J. Intell. Fuzzy Syst., 27(4) (2014), 1621-1629. · Zbl 1305.40006 |
| [29] | Savaş E., Gürdal M., Generalized statistically convergent sequences of functions in fuzzy 2-normed spaces, J. Intell. Fuzzy Syst., 27(4) (2014), 2067-2075. · Zbl 1305.40007 |
| [30] | Savaş, E., Gürdal, M., Ideal convergent function sequences in random 2-normed spaces, Filomat, 30(3) (2016), 557-567. · Zbl 1455.40005 |
| [31] | Savaş, E., Yamancı, U., Gürdal, M., I-lacunary statistical convergence of weighted g via modulus functions in 2-normed spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2) (2019), 2324-2332. · Zbl 1502.40004 |
| [32] | Savaş, R., λ-double statistical convergence of functions, Filomat, 33(2) (2019), 519-524. · Zbl 1500.40005 |
| [33] | Savaş, R., Öztürk, M., On generalized ideal asymptotically statistical equivalent of order α for functions, Ukr. Math. J., 70 (2019), 1901-1912. · Zbl 1458.40008 |
| [34] | Savaş, R.,, Sezer, S. A., Tauberian theorems for sequences in 2-normed spaces, Results Math., 72 (2017), 1919-1931. · Zbl 1387.40003 |
| [35] | Savaş, R. , Patterson, R. F., I₂-lacunary strongly summability for multidimensional measurable functions, Pub. I. Math-Beograd, 107(121) (2020), 93-107. · Zbl 1474.40023 |
| [36] | Şahiner, A., Gürdal, M., Yiğit, T., Ideal convergence characterization of the completion of linear n-normed spaces, Comput. Math. Appl., 61(3) (2011), 683-689. · Zbl 1217.40005 |
| [37] | Siddiqi, A. H., 2-normed spaces, Aligarh Bull. Math., (1980), 53-70. · Zbl 0499.46010 |
| [38] | Vulich, B., On a generalized notion of convergence in a Banach space, Ann. Math., 38(1) (1937), 156-174. · Zbl 0016.06303 |
| [39] | Yamancı, U., Gürdal, M., Statistical convergence and operators on Fock space, New York J. Math., 22 (2016), 199-207. · Zbl 1351.40007 |
| [40] | Yamancı, U., Nabiev, A. A., Gürdal, M., Statistically localized sequences in 2-normed spaces, Honam Math. J., 42(1) (2020), 161-173. Yamancı, U., Savaş, E., Gürdal, M., I-localized sequences in two normed spaces, Malays. J. Math. Sci., 14(3) (2020), 491-503. · Zbl 1468.40002 |
| [41] | Yegül, S., Dündar, E., On statistical convergence of sequences of functions in 2-normed spaces, J. Classical Anal., 10(1) (2017), 49-57. · Zbl 1424.40019 |
| [42] | Yegül, S., Dündar, E., Statistical convergence of double sequences of functions and some properties in 2-normed spaces, Facta Univ. Ser. Math. Inform., 33(5) (2018), 705-719. · Zbl 1474.40007 |
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.
