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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 219435, 16 pages
http://dx.doi.org/10.1155/2012/219435
Research Article

Generalized Stability of Euler-Lagrange Quadratic Functional Equation

Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Republic of Korea

Received 7 May 2012; Accepted 15 July 2012

Academic Editor: Nicole Brillouet-Belluot

Copyright © 2012 Hark-Mahn Kim and Min-Young Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics No. 8, Interscience Publishers, New York, NY, USA, 1960.
  2. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, Mass, USA, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. Borelli and G. L. Forti, “On a general Hyers-Ulam stability result,” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 2, pp. 229–236, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. View at Publisher · View at Google Scholar
  10. G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and their Applications 34, Birkhäuser, Boston, Mass, USA, 1998. View at Publisher · View at Google Scholar
  12. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001.
  13. Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. M. Rassias and H.-M. Kim, “Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 302–309, 2009. View at Publisher · View at Google Scholar
  15. A. Najati and M. B. Moghimi, “Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 399–415, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992. View at Google Scholar · View at Zentralblatt MATH
  17. M. E. Gordji and H. Khodaei, “On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations,” Abstract and Applied Analysis, vol. 2009, Article ID 923476, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. K. Jun, H. Kim, and J. Son, “Generalized Hyers-Ulam stability of a quadratic functional equation,” in Functional Equations in Mathematical Analysis, Th. M. Rassias and J. Brzdek, Eds., chapter 12, pp. 153–164, 2011. View at Google Scholar
  19. K.-W. Jun and H.-M. Kim, “Ulam stability problem for generalized A-quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 305, no. 2, pp. 466–476, 2005. View at Publisher · View at Google Scholar
  20. J.-H. Bae and W.-G. Park, “Stability of a cauchy-jensen functional equation in quasi-banach spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 151547, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. M. E. Gordji and H. Khodaei, “Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5629–5643, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. A. Najati and G. Z. Eskandani, “Stability of a mixed additive and cubic functional equation in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 1318–1331, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. A. Najati and F. Moradlou, “Stability of a quadratic functional equation in quasi-Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 45, no. 3, pp. 587–600, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. T. Z. Xu, J. M. Rassias, M. J. Rassias, and W. X. Xu, “A fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spaces,” Journal of Inequalities and Applications, vol. 2010, Article ID 423231, 23 pages, 2010. View at Publisher · View at Google Scholar
  25. L. G. Wang and B. Liu, “The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-β-normed spaces,” Acta Mathematica Sinica (English Series), vol. 26, no. 12, pp. 2335–2348, 2010. View at Publisher · View at Google Scholar