Abstract
In this article, we introduce the concept of \(\varDelta ^{m}-I\)-convergent and \(\varDelta ^{m}-I\) Cauchy sequences in generalized probabilistic n-normed spaces and establish some results relating to this concept. We also study \(\varDelta ^{m}-I^{*}\) convergence in the same space. Statement Probabilistic norm generalizes and unifies different notions of norm, represented by a distance function, rather than a positive real number. Ideal convergence unifies many notions of convergence of sequences. In this article, we have introduced the notion of generalized difference ideal convergent sequences in probabilistic n-normed space, which generalizes and unifies many existing notions. Hence, the results of this article have been established in general setting.
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References
Menger K (1942) Statistical metrices. Proc Nat Acad Sci USA 28:535–537
\(\check{S}\)erstnev AN (1962) Random normed spaces. Problems Completeness. Kazan Gos U\(\check{c}\)en Zap 122:3–20
Alsina C, Schweizer B, Sklar A (1993) On the definition of a probabilistic normed space. Aequ Math 46:91–98
Karakus S (2007) Statistical convergence on probabilistic normed spaces. Math Commun 12:11–23
Schweizer B, Sklar A (1960) Statistical metric spaces. Pac J Math 10:313–334
Gähler S (1969) Untersuchungen uber verallgemenerte \(m\)-metrische Raume I, II, III. Math Nachr 40:165–189
Gähler S (1964) Lineare 2-normietre Raume. Math Nachr 28:1–43
Gunawan H, Mashadi M (2001) On \(n\)-normed spaces. Int J Math Sci 27:631–639
Jebril IH (2009) Complete random \(n\)-normed linear space. World Appl Sci Jour 6(10):1366–1371
Rahmat MR, Noorani MSM (2007) Probabilistic \(n\)-normed spaces. Int J Stat Econ 1(S07):20–30
Golet I (2011) On generalised probabilistic normed spaces. Int Math Forum 6(42):2073–2078
Kostyrko P, Šalát T, Wilczyński W (2000-2001) \(I\)-convergence. Real Anal Exch 26:669–686
Tripathy BC, Hazarika B (2008) \(I\)-convergent sequence spaces associated with multiplier sequence spaces. Math Inequal Appl 11(3):543–548
Tripathy BC, Hazarika B (2009) Paranormed \(I\)-convergent sequence spaces. Math Slovaca 59(4):485–494
Tripathy BC, Hazarika B (2011) \(I\)-convergent sequences spaces defined by Orlicz function. Acta Math Appl Sin (Eng Ser) 27(1):149–154
Tripathy BC, Hazarika B (2011) \(I\)-monotonic and \(I\)-convergent sequences. Kyungpook Math J 51(2):233–239
Tripathy BC, Hazarika B, Choudhary B (2012) Lacunary \(I\)-convergent sequences. Kyungpook Math J 52(4):473–482
Tripathy BC, Dutta AJ (2012) On \(I\)-acceleration convergence of sequences of fuzzy real numbers. Math Model. Anal 17(4):549–557
Tripathy BC, Mahanta S (2010) On \(I\)-acceleration convergence of sequences. J Frank Inst 347:591–598
Sahiner A, Gürdal M, Saltan S, Gunewan H (2007) Ideal convergence in 2-normed spaces. Taiwan J Math II(5):1477–1484
Kizmaz H (1981) On certain sequence spaces. Can Math Bull 24:169–176
Et M, Colok R (1995) On generalized difference sequence spaces. Soochow J Math 21(4):377–386
Tripathy BC, Esi A (2006) A new type of difference sequence spaces. Int J Sci Technol 1:11–14
Gumus HG, Nuray F (2011) \(\varDelta ^{m}\)-ideal convergence. Selcuk J Appl Math 12:101–110
Tripathy BC, Sen M, Nath S (2012) \(I\)-convergence in probabilistic \(n\)-normed space. Soft Comput 16:1021–1027
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Tripathy, B.C., Sen, M. & Nath, S. On Generalized Difference Ideal Convergence in Generalized Probabilistic n-normed Spaces. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 91, 29–34 (2021). https://doi.org/10.1007/s40010-019-00644-1
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DOI: https://doi.org/10.1007/s40010-019-00644-1