Skip to main content
Journal of Inequalities and Applications
Download PDF
  • Research Article
  • Open access
  • Published:

Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional Equations

Journal of Inequalities and Applications volume 2007, Article number: 041820 (2006) Cite this article

Abstract

We prove the generalized Hyers‐Ulam stability of the following functional inequalities:,, in the spirit of the Rassias stability approach for approximately homomorphisms.

[123456789101112131415161718]

References

  1. Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.

    Google Scholar 

  2. Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222

    Article  MathSciNet  Google Scholar 

  3. Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1

    Article  MathSciNet  MATH  Google Scholar 

  4. Rassias ThM: Problem 16; 2, Report of the 27th International Symp. on Functional Equations. Aequationes Mathematicae 1990, 39: 292–293; 309.

    Google Scholar 

  5. Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056X

    Article  MathSciNet  MATH  Google Scholar 

  6. Rassias ThM, Šemrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society 1992,114(4):989–993. 10.1090/S0002-9939-1992-1059634-1

    Article  MathSciNet  MATH  Google Scholar 

  7. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.

    Book  MATH  Google Scholar 

  8. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser Boston, Boston, Mass, USA; 1998:vi+313.

    Google Scholar 

  9. Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126–130. 10.1016/0022-1236(82)90048-9

    Article  MathSciNet  MATH  Google Scholar 

  10. Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211

    Article  MathSciNet  MATH  Google Scholar 

  11. Jun K-W, Lee Y-H: A generalization of the Hyers-Ulam-Rassias stability of the pexiderized quadratic equations. Journal of Mathematical Analysis and Applications 2004,297(1):70–86. 10.1016/j.jmaa.2004.04.009

    Article  MathSciNet  MATH  Google Scholar 

  12. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.

    MATH  Google Scholar 

  13. Park C: Homomorphisms between Poisson-algebras. Bulletin of the Brazilian Mathematical Society. New Series 2005,36(1):79–97. 10.1007/s00574-005-0029-z

    Article  MathSciNet  MATH  Google Scholar 

  14. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. to appear in Bulletin des Sciences MathĂŠmatiques to appear in Bulletin des Sciences MathĂŠmatiques

  15. Gilányi A: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequationes Mathematicae 2001,62(3):303–309. 10.1007/PL00000156

    Article  MathSciNet  MATH  Google Scholar 

  16. Rätz J: On inequalities associated with the Jordan-von Neumann functional equation. Aequationes Mathematicae 2003,66(1–2):191–200. 10.1007/s00010-003-2684-8

    Article  MathSciNet  MATH  Google Scholar 

  17. Gilányi A: On a problem by K. Nikodem. Mathematical Inequalities & Applications 2002,5(4):707–710.

    Article  MathSciNet  MATH  Google Scholar 

  18. Fechner W: Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequationes Mathematicae 2006,71(1–2):149–161. 10.1007/s00010-005-2775-9

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanyang University, Seoul, 133‐791, South Korea

    Choonkil Park

  2. Department of Mathematics, Chungnam National University, Daejeon, 305‐764, South Korea

    Young Sun Cho & Mi-Hyen Han

Authors
  1. Choonkil Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Young Sun Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mi-Hyen Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Choonkil Park.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Park, C., Cho, Y.S. & Han, MH. Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional Equations. J Inequal Appl 2007, 041820 (2006). https://doi.org/10.1155/2007/41820

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/41820

Keywords