Skip to main content
SpringerLink
Log in

On Generalized Difference Ideal Convergence in Generalized Probabilistic n-normed Spaces

  • Research Article
  • Published:
Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Menger K (1942) Statistical metrices. Proc Nat Acad Sci USA 28:535–537

    Article  ADS  Google Scholar 

  2. \(\check{S}\)erstnev AN (1962) Random normed spaces. Problems Completeness. Kazan Gos U\(\check{c}\)en Zap 122:3–20

  3. Alsina C, Schweizer B, Sklar A (1993) On the definition of a probabilistic normed space. Aequ Math 46:91–98

    Article  MathSciNet  Google Scholar 

  4. Karakus S (2007) Statistical convergence on probabilistic normed spaces. Math Commun 12:11–23

    MathSciNet  MATH  Google Scholar 

  5. Schweizer B, Sklar A (1960) Statistical metric spaces. Pac J Math 10:313–334

    Article  MathSciNet  Google Scholar 

  6. Gähler S (1969) Untersuchungen uber verallgemenerte \(m\)-metrische Raume I, II, III. Math Nachr 40:165–189

    Article  Google Scholar 

  7. Gähler S (1964) Lineare 2-normietre Raume. Math Nachr 28:1–43

    Article  MathSciNet  Google Scholar 

  8. Gunawan H, Mashadi M (2001) On \(n\)-normed spaces. Int J Math Sci 27:631–639

    Article  MathSciNet  Google Scholar 

  9. Jebril IH (2009) Complete random \(n\)-normed linear space. World Appl Sci Jour 6(10):1366–1371

    MathSciNet  Google Scholar 

  10. Rahmat MR, Noorani MSM (2007) Probabilistic \(n\)-normed spaces. Int J Stat Econ 1(S07):20–30

    MathSciNet  Google Scholar 

  11. Golet I (2011) On generalised probabilistic normed spaces. Int Math Forum 6(42):2073–2078

    MathSciNet  MATH  Google Scholar 

  12. Kostyrko P, Šalát T, Wilczyński W (2000-2001) \(I\)-convergence. Real Anal Exch 26:669–686

  13. Tripathy BC, Hazarika B (2008) \(I\)-convergent sequence spaces associated with multiplier sequence spaces. Math Inequal Appl 11(3):543–548

    MathSciNet  MATH  Google Scholar 

  14. Tripathy BC, Hazarika B (2009) Paranormed \(I\)-convergent sequence spaces. Math Slovaca 59(4):485–494

    Article  MathSciNet  Google Scholar 

  15. Tripathy BC, Hazarika B (2011) \(I\)-convergent sequences spaces defined by Orlicz function. Acta Math Appl Sin (Eng Ser) 27(1):149–154

    Article  MathSciNet  Google Scholar 

  16. Tripathy BC, Hazarika B (2011) \(I\)-monotonic and \(I\)-convergent sequences. Kyungpook Math J 51(2):233–239

    Article  MathSciNet  Google Scholar 

  17. Tripathy BC, Hazarika B, Choudhary B (2012) Lacunary \(I\)-convergent sequences. Kyungpook Math J 52(4):473–482

    Article  MathSciNet  Google Scholar 

  18. Tripathy BC, Dutta AJ (2012) On \(I\)-acceleration convergence of sequences of fuzzy real numbers. Math Model. Anal 17(4):549–557

    Article  MathSciNet  Google Scholar 

  19. Tripathy BC, Mahanta S (2010) On \(I\)-acceleration convergence of sequences. J Frank Inst 347:591–598

    Article  MathSciNet  Google Scholar 

  20. Sahiner A, Gürdal M, Saltan S, Gunewan H (2007) Ideal convergence in 2-normed spaces. Taiwan J Math II(5):1477–1484

    Article  MathSciNet  Google Scholar 

  21. Kizmaz H (1981) On certain sequence spaces. Can Math Bull 24:169–176

    Article  MathSciNet  Google Scholar 

  22. Et M, Colok R (1995) On generalized difference sequence spaces. Soochow J Math 21(4):377–386

    MathSciNet  Google Scholar 

  23. Tripathy BC, Esi A (2006) A new type of difference sequence spaces. Int J Sci Technol 1:11–14

    Google Scholar 

  24. Gumus HG, Nuray F (2011) \(\varDelta ^{m}\)-ideal convergence. Selcuk J Appl Math 12:101–110

    MathSciNet  MATH  Google Scholar 

  25. Tripathy BC, Sen M, Nath S (2012) \(I\)-convergence in probabilistic \(n\)-normed space. Soft Comput 16:1021–1027

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Tripura University, Suryamaninagar, Agartala, Tripura, 799022, India

    Binod Chandra Tripathy

  2. National Institute of Technology, Silchar, Silchar, Assam, 788010, India

    Mausumi Sen

  3. Department of Mathematics, Silchar Polytechnic, Silchar, Assam, 788015, India

    Soumitra Nath

Authors
  1. Binod Chandra Tripathy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mausumi Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Soumitra Nath
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Binod Chandra Tripathy.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40010-019-00644-1

Keywords

Search

Navigation