Some statistical convergence types of order \(\alpha\) for double set sequences. (English) Zbl 1488.40037
Summary: In this study, we have introduced the concepts of Wijsman statistical convergence of order \(\alpha \), Hausdorff statistical convergence of order \(\alpha\) and Wijsman strongly \(p\)-Cesàro summability of order \(\alpha\) for double set sequences. Also, we have investigated some properties of these concepts and examined the relationships among them.
Keywords:
statistical convergence; Cesàro summability; double sequence; order \(\alpha \); Wijsman convergence; Hausdorff convergence; set sequencesReferences:
| [1] | M. Baronti and P. Papini: Convergence of sequences of sets. In: Methods of Functional Analysis in Approximation Theory, Birkhäuser, Basel, 1986, pp. 133155. · Zbl 0606.54006 |
| [2] | G. Beer: On convergence of closed sets in a metric space and distance functions. Bull. Aust. Math. Soc. 31 (1985), 421432. · Zbl 0558.54007 |
| [3] | G. Beer: Wijsman convergence: A survey. Set-Valued Anal. 2 (1994), 7794. · Zbl 0812.54014 |
| [4] | S. Bhunia, P. Das and S. K. Pal: Restricting statistical convergence. Acta Mathematica Hungarica, 134(1-2) (2012), 153161. · Zbl 1265.40016 |
| [5] | J. S. Connor: The statistical and strongp-Cesàro convergence of sequences. Analysis, 8(1-2), (1988) 4763. · Zbl 0653.40001 |
| [6] | H. Çakall: A study on statistical convergence. Funct. Anal. Approx. Comput. 1(2) (2009), 1924. · Zbl 1265.40033 |
| [7] | H. Çakall and E. Sava³: Statistical convergence of double sequences in topological groups. J. Comput. Anal. Appl. 12(2) (2010), 421426. · Zbl 1204.40006 |
| [8] | R. Çolak: Statistical convergence of orderα. In: Modern Methods in Analysis and Its Applications, Anamaya Publishers, New Delhi, 2010, pp. 121129. |
| [9] | R. Çolak and Ç. A. Bekta³:λ-statistical convergence of orderα. Acta Mathematica Scientia B, 31(3) (2011), 953959. |
| [10] | R. Çolak and Y. Altn: Statistical convergence of double sequences of orderα. Journal of Function Spaces and Applications, 2013(Article ID 682823) (2013), 5 pages. · Zbl 1285.40001 |
| [11] | H. Fast: Sur la convergence statistique. Colloq. Math. 2 (1951), 241244. · Zbl 0044.33605 |
| [12] | J. A. Fridy: On statistical convergence. Analysis, 5(4) (1985), 301313. · Zbl 0588.40001 |
| [13] | A. D. Gadjiev and C. Orhan: Some approximation theorems via statistical convergence. The Rocky Mountain Journal of Mathematics, 32(1) (2001), 129138. · Zbl 1039.41018 |
| [14] | E. Kolk: The statistical convergence in Banach spaces. Acta Comment. Univ. Tartu, 928 (1991), 4152. |
| [15] | I. J. Maddox: Statistical convergence in a locally convex space. Math. Proc. Cambridge Philos. Soc. 104(1) (1988), 141145. · Zbl 0674.40008 |
| [16] | S. A. Mohiuddine, A. Alotaibi and M. Mursaleen: Statistical convergence of double sequences in locally solid Riesz spaces. Abstract and Applied Analysis, 2012(Article ID 719729) (2012), 9 pages. · Zbl 1262.40005 |
| [17] | S. A. Mohiuddine, H. evli and M. Cancan: Statistical convergence of double sequences in fuzzy normed spaces. Filomat, 26(4) (2012), 673681. · Zbl 1289.40027 |
| [18] | F. Móricz: Statistical convergence of multiple sequences. Archiv der Mathematik, 81(1) (2003), 8289. · Zbl 1041.40001 |
| [19] | M. Mursaleen and O. H. H. Edely: Statistical convergence of double sequences. J. Math. Anal. Appl. 288 (2003), 223231. · Zbl 1032.40001 |
| [20] | M. Mursaleen, C. Çakan, S. A. Mohiuddine and E. Sava³: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser. 26(11) (2010), 21312144. · Zbl 1219.40004 |
| [21] | F. Nuray and B. E. Rhoades: Statistical convergence of sequences of sets. Fasc. Math. 49 (2012), 8799. · Zbl 1287.40004 |
| [22] | F. Nuray, U. Ulusu and E. Dündar: Cesàro summability of double sequences of sets. Gen. Math. Notes 25(1) (2014), 818. · Zbl 1379.40004 |
| [23] | F. Nuray, E. Dündar and U. Ulusu: Wijsman statistical convergence of double sequences of sets. Iran. J. Math. Sci. Inform. (in press). · Zbl 1379.40004 |
| [24] | A. Pringsheim: Zur theorie der zweifach unendlichen Zahlenfolgen. Math. Ann. 53 (1900), 289321. · JFM 31.0249.01 |
| [25] | E. Sava³: Some sequence spaces and statistical convergence. Int. J. Math. Math. Sci. 29(5) (2002), 303306. · Zbl 1001.40006 |
| [26] | E. Sava³: Double almost statistical convergence of orderα. Advances in Dierence Equations, 2013(62) (2013), 9 pages. · Zbl 1380.40005 |
| [27] | E. Sava³: OnI-lacunary statistical convergence of orderαfor sequences of sets. Filomat, 29(6) (2015), 12231229. |
| [28] | I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly, 66 (1959), 361375. · Zbl 0089.04002 |
| [29] | Y. Sever, Ö. Talo and B. Altay: On convergence of double sequences of closed sets. Contemporary Analysis and Applied Mathematics, 3(1) (2015), 3049. · Zbl 1357.40008 |
| [30] | H. Steinhaus: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2 (1951), 7374. |
| [31] | T. alát: On statistically convergent sequences of real numbers. Math. Slovaca, 30(2) (1980), 139150. · Zbl 0437.40003 |
| [32] | H. engül and M. Et: OnI-lacunary statistical convergence of orderαof sequences of sets. Filomat, 31(8) (2017), 24032412. |
| [33] | Ö. Talo, Y. Sever and F. Ba³ar: On statistically convergent sequences of closed sets. Filomat, 30(6) (2016), 14971509. · Zbl 1456.40006 |
| [34] | B. C. Tripathy: On statistical convergence. Proc. Est. Acad. Sci. 47(4) (1998), 299 303. · Zbl 0961.40002 |
| [35] | R. A. Wijsman: Convergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70 (1964), 186188 · Zbl 0121.39001 |
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.
