Open Access

Stability Problem of Ulam for Euler-Lagrange Quadratic Mappings

Journal of Inequalities and Applications20082007:010725

Received: 26 May 2007

Accepted: 9 November 2007

Published: 23 January 2008


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


Authors’ Affiliations

Department of Mathematics, College of Natural Sciences, Chungnam National University
Pedagogical Department E. E., National and Capodistrian University of Athens, Section of Mathematics and Informatics


  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.MathSciNetView ArticleMATHGoogle 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-JMathSciNetView ArticleMATHGoogle 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.27.4.222MathSciNetView ArticleMATHGoogle 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/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle 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.1994.1211MathSciNetView ArticleMATHGoogle 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.View ArticleMATHGoogle Scholar
  8. Hyers DH, Rassias ThM: Approximate homomorphisms. Aequationes Mathematicae 1992,44(2–3):125–153.MathSciNetView ArticleMATHGoogle 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.2000.7046MathSciNetView ArticleMATHGoogle Scholar
  10. Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar
  12. Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.MathSciNetMATHGoogle 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.1997.5856MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle 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.2006.11.024MathSciNetView ArticleMATHGoogle 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 LettersGoogle Scholar
  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. 10.1080/10236190600986925MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle 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.MathSciNetGoogle Scholar
  21. Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006,39(3):523–530.MathSciNetMATHGoogle 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.MathSciNetGoogle Scholar
  23. Rassias JM: On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings. Aequationes Mathematicae 2002,64(1–2):62–69.MathSciNetView ArticleMATHGoogle 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.MATHGoogle 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/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
  27. Gilányi A: On the stability of monomial functional equations. Publicationes Mathematicae Debrecen 2000,56(1–2):201–212.MathSciNetMATHGoogle 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.MathSciNetMATHGoogle Scholar


© Hark-Mahn Kim et al. 2007

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