Skip to main content
SpringerLink
Log in

Generalized Statistical Convergence for Sequences of Function in Random 2-Normed Spaces

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 834))

Included in the following conference series:

  • 594 Accesses

Abstract

In this paper, we introduce a new type of convergence for a sequence of function, namely, \(\lambda \)-statistically convergent sequences of functions in random 2-normed space, which is a natural generalization of convergence in random 2-normed space. In particular, following the line of recent work of Karakaya et al. [12], we introduce the concepts of uniform \(\lambda \)-statistical convergence and pointwise \(\lambda \)-statistical convergence in the topology induced by random 2-normed spaces. We define the \(\lambda \)-statistical analog of the Cauchy convergence criterion for pointwise and uniform \(\lambda \)-statistical convergence in a random 2-normed space and give some basic properties of these concepts. In addition, the preservation of continuity by pointwise and uniform \(\lambda \)-statistical convergence is proven.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

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

    Article  MathSciNet  Google Scholar 

  2. Asadollah, A., Nourouz, K.: Convex sets in probabilistic normed spaces. Chaos, Solitons Fractals 36, 322–328 (2008)

    Article  MathSciNet  Google Scholar 

  3. Balcerzak, M., Dems, K., Komisarski, A.: Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 328, 715–729 (2007)

    Article  MathSciNet  Google Scholar 

  4. Caserta, A., Giuseppe, D., Kočinac, L.: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011, 1–11 (2011)

    MathSciNet  MATH  Google Scholar 

  5. Fast, H.: Sur la convergence statistique. Colloquium Math. 2, 241–244 (1951)

    Article  MathSciNet  Google Scholar 

  6. Fridy, J.: On statistical convergence. Analysis 5, 301–314 (1985)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Giuseppe, D., Kočinac, L.: Statistical convergence in topology. Topol. Appl. 156, 28–45 (2008)

    Article  MathSciNet  Google Scholar 

  9. Golet, I.: On probabilistic 2-normed spaces. Novi Sad J. Math. 35, 95–102 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Gürdal, M., Pehlivan, S.: The statistical convergence in 2-banach spaces. Thai J. Math. 2, 107–113 (2004)

    MATH  Google Scholar 

  11. Karakaya, V., Şimşek, N., Ertürk, M., Gürsoy, F.: \(\lambda \)-statistical convergence of sequences of functions with respect to the intuitionistic fuzzy normed spaces. J. Funct. Spaces Appl. 2012, 1–14 (2012)

    Article  Google Scholar 

  12. Karakaya, V., Şimşek, N., Ertürk, M., Gürsoy, F.: Statistical convergence of sequences of functions with respect to the intuitionistic fuzzy normed spaces. Abstr. Appl. Anal. 2012, 1–19 (2012)

    MATH  Google Scholar 

  13. Menger, K.: Statistical metrics. Proc. Nat. Acad. Sci. 28, 535–537 (1942)

    Article  MathSciNet  Google Scholar 

  14. Mursaleen, M.: \(\lambda \)-statistical convergence. Math. Slovaca 50, 111–115 (2000)

    MathSciNet  MATH  Google Scholar 

  15. Mursaleen, M.: On statistical convergence in random 2-normed spaces. Acta Sci. Math. (Szeged) 76, 101–109 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Savaş, E.: On generalized statistical convergence in random 2-normed space. Iran. J. Sci. Technol. A4, 417–423 (2012)

    MathSciNet  MATH  Google Scholar 

  17. Savaş, E., Gürdal, M.: Certain summability methods in intuitionistic fuzzy normed spaces. J. Intell. Fuzzy Syst. 27, 1621–1629 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Savaş, E., Mohiuddine, S.: \(\lambda \)-statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 62, 99–108 (2012)

    Article  MathSciNet  Google Scholar 

  19. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. Elsevier Science Publishing Co., New York (1983)

    MATH  Google Scholar 

  20. Sempi, C.: A short and partial history of probabilistic normed spaces. Mediterr. J. Math. 3, 283–300 (2006)

    Article  MathSciNet  Google Scholar 

  21. Serstnev, A.: On the notion of a random normed space. Dokl. Akad. Nauk SSSR 149, 280–283 (1963)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Istanbul Ticaret University, Üsküdar-Istanbul, Turkey

    Ekrem Savaş

  2. Department of Mathematics, Suleyman Demirel University, 32260, Isparta, Turkey

    Mehmet Gürdal

Authors
  1. Ekrem Savaş
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehmet Gürdal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mehmet Gürdal .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology BHU, Varanasi, Uttar Pradesh, India

    Debdas Ghosh

  2. Haldia Institute of Technology, Haldia, India

    Debasis Giri

  3. University of Central Florida, Orlando, Florida, USA

    Ram N. Mohapatra

  4. Istanbul Commerce University, Istanbul, Turkey

    Ekrem Savas

  5. Kyushu University, Fukuoka, Japan

    Kouichi Sakurai

  6. Indian Institute of Technology (BHU), Varanasi, India

    L. P. Singh

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Savaş, E., Gürdal, M. (2018). Generalized Statistical Convergence for Sequences of Function in Random 2-Normed Spaces. In: Ghosh, D., Giri, D., Mohapatra, R., Savas, E., Sakurai, K., Singh, L. (eds) Mathematics and Computing. ICMC 2018. Communications in Computer and Information Science, vol 834. Springer, Singapore. https://doi.org/10.1007/978-981-13-0023-3_28

Download citation

  • DOI: https://doi.org/10.1007/978-981-13-0023-3_28

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-13-0022-6

  • Online ISBN: 978-981-13-0023-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation