Skip to main content
SpringerLink
Log in

Superstability of Generalized Module Left Higher Derivations on a Multi-Banach Module

  • Chapter
  • First Online:
Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 96))

  • 1544 Accesses

Abstract

The problem of stability of functional equations was originally raised by Ulam in 1940. During the last decades, several stability problems for various functional equations have been investigated by several authors. In this chapter, by defining a multi-Banach space, we introduce a multi-Banach module. Also, we define the notion of generalized module left higher derivations and approximate generalized module left higher derivations. Then, we discuss the superstability of an approximate generalized module left higher derivation on a multi-Banach module. In fact, we show that an approximate generalized module left higher derivation on a multi-Banach module is a generalized module left higher derivation. Finally, we get the similar result for a linear generalized module left higher derivation.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Similar content being viewed by others

References

  1. Aoki, T.: On the stability of the linear transformations in Banach spaces. J. Math. Soc. Jpn.2, 64–66 (1950)

    Article  MATH  Google Scholar 

  2. Baak, C., Boo, D.-H., Rassias, Th.M.: Generalized additive mapping in Banach modules and isomorphisms between C* -algebras. J. Math. Anal. Appl.314(1), 150–161 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bland, P.E.: Higher derivations on rings and modules. Int. J. Math. Math. Sci.15, 2373–2387 (2005)

    Article  MathSciNet  Google Scholar 

  4. Bourgin, D.G.: Classes of transformations and bordering transformations. Bull. Am. Math. Soc.57, 223–237 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cao, H.-X., Lv, J.-R., Rassias, J.M.: Superstability for generalized module left derivations and generalized module derivations on a Banach module \((II)\). J. Inequal. Pure Appl. Math.10(2), Art. 8–5 (2009)

    MathSciNet  Google Scholar 

  6. Czerwik, S.: Stability of Functional Equations of Ulam–Hyers–Rassias Type. Hadronic, Palm Harbor (2003)

    Google Scholar 

  7. Dales, H.G., Polyakov, M.E.: Multi-normed spaces and multi-Banach algebras. (preprint)

    Google Scholar 

  8. Hasse, H., Schmidt, F.K.: Noch eine Begrüdung der theorie der höheren differential quotienten in einem algebraischen funtionenkörper einer unbestimmeten. J. Reine Angew. Math.177, 215–237 (1937)

    Google Scholar 

  9. Hejazian, S., Shatery, T.L.: Automatic continuity of higher derivations on \(JB*\)-algebras. Bull. Iran. Math. Soc.33(1), 11–23 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Hejazian, S., Shatery, T.L.: Higher derivations on Banach algebras. J. Anal. Appl.6, 1–15 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hejazian, S., Shateri, T.L.: [AQ1]A characterization of higher derivations, to appear in Italian. J. Pure Appl. Math. (2015)

    Google Scholar 

  12. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A.27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  13. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  14. Jewell, N.P.: Continuity of module and higher derivations. Pac. J. Math.68, 91–98 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)

    MATH  Google Scholar 

  16. Jung, S.-M., Popa, D., Rassias, M.Th: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. (to appear)

    Google Scholar 

  17. Kang, S.-Y., Chang, I.-S.: Approximation of generalized left derivations. J. Abst. Appl. Anal.2008, 1–8 (2008)

    Article  MathSciNet  Google Scholar 

  18. Lee, Y.-H., Jung, S.-M., Rassias, M.Th: On an n-dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput. (to appear)

    Google Scholar 

  19. Park, C.-G.: Linear derivations on Banach algebras. Nonlinear Funct. Anal. Appl.9, 359–368 (2004)

    MATH  MathSciNet  Google Scholar 

  20. Park, C.-G.: Lie *-homomorphisms between Lie C*-algebras and Lie *- derivations on Lie C*-algebras. J. Math. Anal. Appl.293, 419–434 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Park, C.-G., Rassias, Th.M.: Hyers-Ulam stability of a generalized Apollonius type quadratic mapping. J. Math. Anal. Appl.322(1), 371–381 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rassias, Th.M.: On the stability of the linear mappings in Banach space. Proc. Am. Math. Soc.72, 297–300 (1978)

    Article  MATH  Google Scholar 

  23. Rassias, Th.M.: Functional Equations, Inequalities and Applications. Kluwer, Dordrecht (2003)

    Book  MATH  Google Scholar 

  24. Rassias, Th.M., Tabor, J.: Stability of Mappings of Hyers - Ulam Type. Hadronic Press, Florida (1994)

    MATH  Google Scholar 

  25. Shateri, T.L.: Superstability of generalized higher derivations. Abstr. Appl. Anal. (2011). doi:10.1155/2011/239849

    MathSciNet  Google Scholar 

  26. Ulam, S.M.: A Collection of Mathematical Problems. Problems in Modern Mathematics. Wiley, New York (1964)

    Google Scholar 

  27. Uchino, Y., Satoh, T.: Function field modular forms and higher derivations. Math. Ann.311, 439–466 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, P. O. Box 397, Sabzevar, Iran

    T. L. Shateri & Z. Afshari

Authors
  1. T. L. Shateri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Z. Afshari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. L. Shateri .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Shateri, T., Afshari, Z. (2014). Superstability of Generalized Module Left Higher Derivations on a Multi-Banach Module. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 96. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1286-5_16

Download citation

Publish with us

Policies and ethics

Search

Navigation