On the Hyers–Ulam Stability of Functional Equations with Respect to Bounded Distributions

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)

Abstract

We consider the Hyers–Ulam stability of the Cauchy, Jensen, Pexider, Pexider–Jensen equations with respect to bounded distributions. We also consider the Hyers–Ulam–Rassias stability problem for the quadratic functional equation in the space of Fourier hyperfunctions.

Keywords

Bounded distribution Fourier hyperfunction Cauchy equation Pexider equation Jensen equation Quadratic functional equation Heat kernel Hyers–Ulam stability 

References

  1. 1.
    Aczél, J., Dhombres, J.: Functional equations in several variables. Cambridge University Press, New York-Sydney (1989)MATHCrossRefGoogle Scholar
  2. 2.
    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Baker, J.A.: On a functional equation of Aczél and Chung. Aequationes Math. 46, 99–111 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Baker, J.A.: The stability of cosine functional equation. Proc. Amer. Math. Soc. 80, 411–416 (1980)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bourgin, D.G.: Class of transformations and bordering transformations. Bull. Amer. Math. Soc. 57, 223–237 (1951)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bourgin, D.G.: Multiplicative transformations. Proc. Nat. Academy Sci. U.S.A. 36, 564–570 (1950)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chung, J.: Distributional method for a class of functiuonal equations and their stabilities. Acta Math. Sinica 23, 2017–2026 (2007)MATHCrossRefGoogle Scholar
  8. 8.
    Chung, J.: Stability of approximately quadratic Schwartz distributions. Nonlinear Anal. 67, 175–186 (2007)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chung, J.: A distributional version of functional equations and their stabilities. Nonlinear Anal. 62, 1037–1051 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Chung, J., Chung, S.-Y., Kim, D.: A characterization for Fourier hyperfunctions. Publ. Res. Inst. Math. Sci. 30, 203–208 (1994)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chung, S.-Y., Kim, D., Lee, E.G.: Periodic hyperfunctions and Fourier series. Proc. Amer. math. Soc. 128, 2421–2430 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Czerwik, S.: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Inc., Palm Harbor (2003)Google Scholar
  13. 13.
    Deeba, E.Y., Koh, E.L.: The Pexider functional equations in distributions. Canad. J. Math. 42, 304–314 (1990)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Deeba, E., Xie, S., Distributional analog of a functional equation. Appl. Math. Lett. 16, 669–673 (2003)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fenyö, I.: Über eine Losungsmethode gewisser Funktionalgleichungen. Acta Math. Acad. Sci. Hungar. 7, 383–396 (1956)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Forti, G.L.: Hyer-Ulam stability of functional equation in several variables. Aeqationes Math. 50, 143–190 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Forti, G.L.: The stability of homomorphisms and amenablity with applications to functional equations. Abh. Math. Sem. Univ. Hamburg 57, 215–226 (1987)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Gelfand, I.M., Shilov, G.E.: Generalized Functions II. Academic Press, New York (1968)Google Scholar
  19. 19.
    Hörmander, L.: The Analysis of Linear Partial Differential Operator I. Springer–Verlag, Berlin–New York (1983)Google Scholar
  20. 20.
    Hyers, D.H.: On the stability of the linear functional equations. Proc. Nat. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequationes Math. 44, 125–153 (1992)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhauser (1998)Google Scholar
  23. 23.
    Jung, S.M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Inc., Palm Harbor (2001)MATHGoogle Scholar
  24. 24.
    Jun, K.W., Kim, H.M.: Stability problem for Jensen-type functional equations of cubic mappings, Acta Math. Sinica, English Series, 22(6), 1781–1788 (2006)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kim, G.H., Lee, Y.H.: Boundedness of approximate trigonometric functional equations. Appl. Math. Lett. 31, 439–443 (2009)CrossRefGoogle Scholar
  26. 26.
    Matsuzawa, T.: A calculus approach to hyperfunctions III. Nagoya Math. J. 118, 133–153 (1990)MathSciNetMATHGoogle Scholar
  27. 27.
    Park, C.G.: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algabras. Bull. Sci. Math. 132, 87–96 (2008)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Rassias, J.M., Rassias, M.J.: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281, 516–524 (2003)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rassias, J.M.: On Approximation of Approximately Linear Mappings by Linear Mappings. J. Funct. Anal. 46, 126–130 (1982)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rassias, Th.M.: On the stability of linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1966)MATHGoogle Scholar
  33. 33.
    Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publ., New York (1960)MATHGoogle Scholar
  34. 34.
    Widder, D.V.: The Heat Equation. Academic Press, New York (1975)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsKunsan National UniversityKunsanKorea

Personalised recommendations