Advertisement

Acta Mathematica Sinica

, Volume 22, Issue 6, pp 1789–1796 | Cite as

Cauchy–Rassias Stability of Cauchy–Jensen Additive Mappings in Banach Spaces

  • Choonkil Baak
ORIGINAL ARTICLES

Abstract

Let X, Y be vector spaces. It is shown that if a mapping f : XY satisfies
$$ f{\left( {\frac{{x + y}} {2} + z} \right)} + f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( x \right)} + 2f{\left( z \right)}, $$
(0.1)
$$ f{\left( {\frac{{x + y}} {2} + z} \right)} - f{\left( {\frac{{x - y}} {2} + z} \right)} = f{\left( y \right)}, $$
(0.2) or
$$ 2f{\left( {\frac{{x + y}} {2} + z} \right)} = f{\left( x \right)} + f{\left( y \right)} + 2f{\left( z \right)} $$
(0.3) for all x, y, zX, then the mapping f : XY is Cauchy additive. Furthermore, we prove the Cauchy–Rassias stability of the functional equations (0.1), (0.2) and (0.3) in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.

Keywords

Cauchy additive mapping Jensen additive mapping Cauchy–Rassias stability isomorphism between Banach algebra 

MR (2000) Subject Classification

39B52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ulam, S. M.: Problems in Modern Mathematics, Wiley, New York, 1960Google Scholar
  2. 2.
    Hyers, D. H.: On the stability of the linear functional equation. Pro. Nat’l. Acad. Sci. U.S.A., 27, 222–224 (1941)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Rassias, Th. M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc., 72, 297–300 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Haruki, H., Rassias, Th. M.: New generalizations of Jensen’s functional equation. Proc. Amer. Math. Soc., 123, 495–503 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hou, J., Park, C.: Homomorphisms between C*–algebras associated with the Trif functional equation and linear derivations on C*–algebras. J. Korean Math. Soc., 41, 461–477 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Park, C.: On the stability of the linear mapping in Banach modules. J. Math. Anal. Appl.,275, 711–720 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Park, C.: Universal Jensen’s equations in Banach modules over a C*–algebra and its unitary group. Acta Math. Sinica, English Series, 20, 1047–1056 (2004)zbMATHCrossRefGoogle Scholar
  8. 8.
    Park, C., Hou, J., Oh, S.: Homomorphisms between JC*–algebras and between Lie C*–algebras. Acta Math. Sinica, English Series, 21(6), 1391–1398 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Park, C., Park, J., Shin, J.: Hyers–Ulam–Rassias stability of quadratic functional equations in Banach modules over a C*–algebra. Chinese Ann. Math., Series B, 24, 261–266 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    Rassias, J. M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal., 46, 126–130 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Rassias, J. M.: Solution of a problem of Ulam. J. Approx. Theory, 57, 268–273 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Rassias, Th. M.: On a modified Hyers–Ulam sequence. J. Math. Anal. Appl., 158, 106–113 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Rassias, Th. M.: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl., 246, 352–378 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Rassias, Th. M.: On the stability of functional equations in Banach spaces.J. Math. Anal. Appl. , 251, 264–284 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Th.M. Rassias: On the stability of functional equations and a problem of Ulam. Acta Appl. Math., 62, 23–130 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Rassias, Th. M.,Šemrl, P.: On the behavior of mappings which do not satisfy Hyers–Ulam stability. Proc. Amer. Math. Soc., 114, 989–993 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Rassias, Th. M., Šemrl, P.: On the Hyers–Ulam stability of linear mappings. J. Math. Anal. Appl., 173, 325–338 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Rassias, Th. M., Shibata, K.: Variational problem of some quadratic functionals in complex analysis. J. Math. Anal. Appl., 228, 234–253 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Găvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl., 184, 431–436 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jun, K., Lee, Y.: A generalization of the Hyers–Ulam–Rassias stability of Jensen’s equation. J. Math. Anal. Appl., 238, 305–315 (1999)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsChungnam National UniversityDaejeon 305–764South Korea

Personalised recommendations