Advertisement

Mediterranean Journal of Mathematics

, Volume 6, Issue 2, pp 233–248 | Cite as

Hyers–Ulam–Rassias Stability of a Quadratic and Additive Functional Equation in Quasi-Banach Spaces

  • Fridoun Moradlou
  • Hamid Vaezi
  • G. Zamani Eskandani
Article

Abstract

In this paper we establish the general solution of the functional equation
$$f(x + 2y) + f(x - 2y) + 4f(x) = 3[f(x + y) + f(x - y)] + f(2y) - 2f(y)$$
and investigate the Hyers–Ulam–Rassias stability of this equation in quasi-Banach spaces. The concept of Hyers–Ulam–Rassias stability originated from Th. M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

Mathematics Subject Classification (2000)

Primary 39B52 Secondary 39B72 47Jxx 

Keywords

Hyers–Ulam–Rassias stability quadratic function additive function quasi-Banach space p-Banach space 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aczél J.: Short Course on Functional Equations. D. Reidel Publishing Co., Dordrecht (1987)MATHGoogle Scholar
  2. 2.
    Aczél J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)MATHGoogle Scholar
  3. 3.
    Amir D.: Characterizations of Inner Product Spaces. Birkhäuser, Basel (1986)MATHGoogle Scholar
  4. 4.
    Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, American Mathematical Society Colloquium Publications 48, American Mathematical Society, Providence, RI, 2000.Google Scholar
  5. 5.
    Cholewa P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62, 59–64 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eskandani G.Z.: On the Hyers-Ulam-Rassias stability of an additive functional equation in quasi-Banach spaces. J. Math. Anal. Appl. 345, 405–409 (2008)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Găvruta P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grabiec A.: The generalized Hyers-Ulam stability of a class of functional equations. Publ. Math. Debrecen 48, 217–235 (1996)MathSciNetGoogle Scholar
  10. 10.
    Hyers D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27, 222–224 (1941)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Jordan P., Von Neumann J.: On inner products in linear metric spaces. Ann. of Math. 36, 719–723 (1935)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Jun K., Lee Y.: On the Hyers–Ulam–Rassias stability of a Pexiderized quadratic inequality. Math. Inequal. Appl. 4, 93–118 (2001)MATHMathSciNetGoogle Scholar
  13. 13.
    Kannappan Pl.: Quadratic functional equation and inner product spaces. Results Mat. 27, 368–372 (1995)MathSciNetGoogle Scholar
  14. 14.
    F. Moradlou, H. Vaezi and C. Park, Fixed points and stability of an additive functional equation of n-Apollonius type in C * -algebras, Abstr. Appl. Anal. (2008), Art. ID 672618.Google Scholar
  15. 15.
    M.S. Moslehian, On the orthogonal stability of the Pexiderized quadratic equation, J. Difference Equ. Appl. 11, No.11 (2005), 999–1004Google Scholar
  16. 16.
    Najati A., Moghimi M.B.: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. J. Math. Anal. Appl. 337, 399–415 (2008)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Paganoni L., Rătz J.: Conditional function equations and orthogonal additivity. Aequationes Math. 50, 135–142 (1995)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Rassias Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rassias Th.M., Tabor J.: What is left of Hyers-Ulam stability?. J. Natur. Geom. 1, 65–69 (1992)MATHMathSciNetGoogle Scholar
  20. 20.
    S. Rolewicz, Metric Linear Spaces, PWN–Polish Scientific Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1984.Google Scholar
  21. 21.
    Skof F.: Local properties and approximations of operators. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S.M. Ulam, A Collection of the Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics 8, Interscience Publishers, New York–London, 1960.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Fridoun Moradlou
    • 1
  • Hamid Vaezi
    • 2
  • G. Zamani Eskandani
    • 2
  1. 1.Department of MathematicsSahand University of TechnologyTabrizIran
  2. 2.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

Personalised recommendations