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Abstract and Applied Analysis
Volume 2011, Article ID 502903, 15 pages
http://dx.doi.org/10.1155/2011/502903
Research Article

Stability in Generalized Functions

Department of Mathematics, Sogang University, Seoul 121-741, Republic of Korea

Received 31 July 2011; Revised 17 September 2011; Accepted 21 September 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 Young-Su Lee. 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. D. H. Hyers, β€œOn 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 Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, NY, USA, 1964.
  3. G. L. Forti, β€œThe stability of homomorphisms and amenability, with applications to functional equations,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 57, pp. 215–226, 1987. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  4. T. Aoki, β€œOn the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. T. M. Rassias, β€œOn 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 Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. Z. Gajda, β€œOn stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. G. Isac and T. M. Rassias, β€œOn the Hyers-Ulam stability of ψ-additive mappings,” Journal of Approximation Theory, vol. 72, no. 2, pp. 131–137, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. L. P. Castro and A. Ramos, β€œHyers-Ulam-Rassias stability for a class of nonlinear Volterra integral equations,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 36–43, 2009. View at Google Scholar
  9. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing, River Edge, NJ, USA, 2002. View at Publisher Β· View at Google Scholar
  10. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.
  11. K.-W. Jun and H.-M. Kim, β€œOn the stability of an n-dimensional quadratic and additive functional equation,” Mathematical Inequalities & Applications, vol. 9, no. 1, pp. 153–165, 2006. View at Google Scholar
  12. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.
  13. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2009. View at Publisher Β· View at Google Scholar
  14. G. H. Kim, β€œStability of the pexiderized lobacevski equation,” Journal of Applied Mathematics, vol. 2011, Article ID 540274, 10 pages, 2011. View at Publisher Β· View at Google Scholar
  15. M. S. Moslehian and D. Popa, β€œOn the stability of the first-order linear recurrence in topological vector spaces,” Nonlinear Analysis, vol. 73, no. 9, pp. 2792–2799, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. B. Paneah, β€œSome remarks on stability and solvability of linear functional equations,” Banach Journal of Mathematical Analysis, vol. 1, no. 1, pp. 56–65, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  17. T. Trif, β€œOn the stability of a general gamma-type functional equation,” Publicationes Mathematicae Debrecen, vol. 60, no. 1-2, pp. 47–61, 2002. View at Google Scholar Β· View at Zentralblatt MATH
  18. P. Nakmahachalasint, β€œOn the Hyers-Ulam-Rassias stability of an n-dimensional additive functional equation,” Thai Journal of Mathematics, vol. 5, no. 3, pp. 81–86, 2007. View at Google Scholar
  19. J. Chung and S. Lee, β€œSome functional equations in the spaces of generalized functions,” Aequationes Mathematicae, vol. 65, no. 3, pp. 267–279, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  20. J. Chung, S.-Y. Chung, and D. Kim, β€œThe stability of Cauchy equations in the space of Schwartz distributions,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 107–114, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  21. J. Chung, β€œA distributional version of functional equations and their stabilities,” Nonlinear Analysis, vol. 62, no. 6, pp. 1037–1051, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  22. Y.-S. Lee and S.-Y. Chung, β€œThe stability of a general quadratic functional equation in distributions,” Publicationes Mathematicae Debrecen, vol. 74, no. 3-4, pp. 293–306, 2009. View at Google Scholar Β· View at Zentralblatt MATH
  23. Y.-S. Lee and S.-Y. Chung, β€œStability of quartic functional equations in the spaces of generalized functions,” Advances in Difference Equations, vol. 2009, Article ID 838347, 16 pages, 2009. View at Google Scholar Β· View at Zentralblatt MATH
  24. L. Hörmander, The Analysis of Linear Partial Differential Operators I, vol. 256, Springer, Berlin, Germany, 1983.
  25. L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966.
  26. J. Chung, S.-Y. Chung, and D. Kim, β€œA characterization for fourier hyperfunctions,” Publications of the Research Institute for Mathematical Sciences, vol. 30, no. 2, pp. 203–208, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  27. T. Matsuzawa, β€œA calculus approach to hyperfunctions. III,” Nagoya Mathematical Journal, vol. 118, pp. 133–153, 1990. View at Google Scholar Β· View at Zentralblatt MATH
  28. K. W. Kim, S.-Y. Chung, and D. Kim, β€œFourier hyperfunctions as the boundary values of smooth solutions of heat equations,” Publications of the Research Institute for Mathematical Sciences, vol. 29, no. 2, pp. 289–300, 1993. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  29. S.-M. Jung, β€œHyers-Ulam-Rassias stability of Jensen's equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137–3143, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet