Skip to main content
  • Research Article
  • Open access
  • Published:

Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings

Abstract

We solve the generalized Hyers-Ulam stability problem for multidimensional Euler-Lagrange quadratic mappings which extend the original Euler-Lagrange quadratic mappings.

[12345678910111213141516171819202122232425262728]

References

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

    Google Scholar 

  2. Gruber PM: Stability of isometries. Transactions of the American Mathematical Society 1978, 245: 263–277.

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhou DX: On a conjecture of Z. Ditzian. Journal of Approximation Theory 1992,69(2):167–172. 10.1016/0021-9045(92)90140-J

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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 

  6. 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 

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

    Book  MATH  Google Scholar 

  8. Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153.

    Article  MathSciNet  MATH  Google Scholar 

  9. Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046

    Article  MathSciNet  MATH  Google Scholar 

  10. Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.

    MathSciNet  MATH  Google Scholar 

  11. Rassias JM: On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces. Journal of Mathematical and Physical Sciences 1994,28(5):231–235.

    MathSciNet  MATH  Google Scholar 

  12. Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.

    MathSciNet  MATH  Google Scholar 

  13. Rassias JM: Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings. Journal of Mathematical Analysis and Applications 1998,220(2):613–639. 10.1006/jmaa.1997.5856

    Article  MathSciNet  MATH  Google Scholar 

  14. Rassias JM: On the stability of the multi-dimensional Euler-Lagrange functional equation. The Journal of the Indian Mathematical Society 1999,66(1–4):1–9.

    MathSciNet  MATH  Google Scholar 

  15. Rassias MJ, Rassias JM: On the Ulam stability for Euler-Lagrange type quadratic functional equations. The Australian Journal of Mathematical Analysis and Applications 2005,2(1, article 11):1–10.

    MathSciNet  MATH  Google Scholar 

  16. Jun K-W, Kim H-M: On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2007,332(2):1335–1350. 10.1016/j.jmaa.2006.11.024

    Article  MathSciNet  MATH  Google Scholar 

  17. Lee Y-S, Chung S-Y: Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions. to appear in Applied Mathematics Letters to appear in Applied Mathematics Letters

  18. Park C, Park JM: Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping. Journal of Difference Equations and Applications 2006,12(12):1277–1288. 10.1080/10236190600986925

    Article  MathSciNet  MATH  Google Scholar 

  19. Park C-G: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between-algebras. Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619–632.

    MathSciNet  MATH  Google Scholar 

  20. Park C-G, Rassias JM: Hyers-Ulam stability of an Euler-Lagrange type additive mapping. International Journal of Applied Mathematics & Statistics 2007,7(Fe07):112–125.

    MathSciNet  Google Scholar 

  21. Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006,39(3):523–530.

    MathSciNet  MATH  Google Scholar 

  22. Ravi K, Arunkumar M: On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation. International Journal of Applied Mathematics & Statistics 2007,7(Fe07):143–156.

    MathSciNet  Google Scholar 

  23. Rassias JM: On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings. Aequationes Mathematicae 2002,64(1–2):62–69.

    Article  MathSciNet  MATH  Google Scholar 

  24. Benyamini Y, Lindenstrauss J: Geometric Nonlinear Functional Analysis. Vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, USA; 2000:xii+488.

    MATH  Google Scholar 

  25. Rolewicz S: Metric Linear Spaces. 2nd edition. PWN-Polish Scientific, Warsaw, Poland; 1984:xi+459.

    Google Scholar 

  26. 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 

  27. Gilányi A: On the stability of monomial functional equations. Publicationes Mathematicae Debrecen 2000,56(1–2):201–212.

    MathSciNet  MATH  Google Scholar 

  28. Kurepa S: On the quadratic functional. Publications de l'Institut Mathématique de l'Académie Serbe des Sciences et des Arts 1961, 13: 57–72.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hark-Mahn Kim.

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

Kim, HM., Rassias, J.M. & Cho, YS. Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings. J Inequal Appl 2007, 010725 (2008). https://doi.org/10.1155/2007/10725

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/10725

Keywords