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Abstract and Applied Analysis
Volume 2007, Article ID 35151, 15 pages
http://dx.doi.org/10.1155/2007/35151
Research Article

On the Generalized Hyers-Ulam Stability of a Cauchy-Jensen Functional Equation

1National Institute for Mathematical Sciences, Daejeon 305-340, South Korea
2Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea
3Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, South Korea

Received 2 July 2007; Revised 27 August 2007; Accepted 22 October 2007

Academic Editor: John Michael Rassias

Copyright © 2007 Kil-Woung Jun et al. 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 Mathematical Problems, Interscience, New York , NY, USA, 1968.
  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, no. 4, pp. 222–224, 1941. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. P. M. Gruber, “Stability of isometries,” Transactions of the American Mathematical Society, vol. 245, pp. 263–277, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order,” Applied Mathematics Letters, vol. 17, no. 10, pp. 1135–1140, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H.-M. Kim, “On the stability problem for a mixed type of quartic and quadratic functional equation,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 358–372, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Y.-H. Lee and K.-W. Jun, “On the stability of approximately additive mappings,” Proceedings of the American Mathematical Society, vol. 128, no. 5, pp. 1361–1369, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H.-M. Kim and J. M. Rassias, “Generalization of Ulam stability problem for Euler-Lagrange quadratic mappings,” Journal of Mathematical Analysis and Applications , vol. 336, no. 1, pp. 277–296, 2007. View at Publisher · View at Google Scholar
  11. C.-G. Park, “Linear functional equations in Banach modules over a C-algebra,” Acta Applicandae Mathematicae, vol. 77, no. 2, pp. 125–161, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. M. Rassias, “Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 613–639, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. J. M. Rassias, “On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings,” Aequationes Mathematicae, vol. 64, no. 1-2, pp. 62–69, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. M. Rassias and M. J. Rassias, “Asymptotic behavior of alternative Jensen and Jensen type functional equations,” Bulletin des Sciences Mathématiques, vol. 129, no. 7, pp. 545–558, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. M. Rassias, “Solution of a Cauchy-Jensen stability Ulam type problem,” Archivum Mathematicum, vol. 37, no. 3, pp. 161–177, 2001. View at Google Scholar · View at MathSciNet
  18. K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,” to appear in Journal of Inequalities in Pure and Applied Mathematics.
  19. W.-G. Park and J.-H. Bae, “On a Cauchy-Jensen functional equation and its stability,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 634–643, 2006. View at Publisher · View at Google Scholar · View at MathSciNet