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Abstract and Applied Analysis
Volume 2013, Article ID 275915, 12 pages
http://dx.doi.org/10.1155/2013/275915
Research Article

On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

Jaeyoung Chung1 and Jeongwook Chang2

1Department of Mathematics, Kunsan National University, Kunsan 573-701, Republic of Korea
2Department of Mathematics Education, Dankook University, Yongin 448-701, Republic of Korea

Received 6 February 2013; Accepted 10 April 2013

Academic Editor: Adem Kılıçman

Copyright © 2013 Jaeyoung Chung and Jeongwook Chang. 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 Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York, NY, USA, 1960. View at MathSciNet
  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 · View at MathSciNet
  3. T. Aoki, “On 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. G. Bourgin, “Multiplicative transformations,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 564–570, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. G. Bourgin, “Approximately isometric and multiplicative transformations on continuous function rings,” Duke Mathematical Journal, vol. 16, pp. 385–397, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Th. 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
  7. Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. A. Baker, “On a functional equation of Aczél and Chung,” Aequationes Mathematicae, vol. 46, no. 1-2, pp. 99–111, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. A. Baker, “The stability of the cosine equation,” Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Székelyhidi, “The stability of d'Alembert-type functional equations,” Acta Scientiarum Mathematicarum, vol. 44, no. 3-4, pp. 313–320, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Chung, “A distributional version of functional equations and their stabilities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 62, no. 6, pp. 1037–1051, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Chang and J. Chung, “Stability of trigonometric functional equations in generalized functions,” Journal of Inequalities and Applications, vol. 2012, Article ID 801502, 12 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Palm Harbor, Fla, USA, 2003.
  14. 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 · View at MathSciNet
  15. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser Boston Publisher, Boston, Mass, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  16. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. View at MathSciNet
  17. K.-W. Jun and H.-M. Kim, “Stability problem for Jensen-type functional equations of cubic mappings,” Acta Mathematica Sinica (English Series), vol. 22, no. 6, pp. 1781–1788, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. G. H. Kim, “On the stability of the Pexiderized trigonometric functional equation,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 99–105, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. G. H. Kim and Y. H. Lee, “Boundedness of approximate trigonometric functional equations,” Applied Mathematics Letters, vol. 31, pp. 439–443, 2009. View at Google Scholar
  20. C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87–96, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. 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
  22. J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques. 2e Série, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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
  24. L. Székelyhidi, “The stability of the sine and cosine functional equations,” Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 109–115, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. I. Tyrala, “The stability of d'Alembert's functional equation,” Aequationes Mathematicae, vol. 69, no. 3, pp. 250–256, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 2. Spaces of Fundamental and Generalized Functions, Academic Press, New York, NY, USA, 1968. View at MathSciNet
  27. L. Hörmander, The Analysis of Linear Partial Differential Operators. I, vol. 256 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  28. T. Matsuzawa, “A calculus approach to hyperfunctions. III,” Nagoya Mathematical Journal, vol. 118, pp. 133–153, 1990. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966. View at MathSciNet
  30. D. V. Widder, The Heat Equation. Pure and Applied Mathematics, vol. 67, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  31. J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. I. Fenyö, “Über eine Lösungsmethode gewisser Funktionalgleichungen,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 7, pp. 383–396, 1956. View at Publisher · View at Google Scholar · View at MathSciNet