Real Analysis Exchange

Stability of Two Types of Cubic Functional Equations in Non-Archimedean Spaces

Mohammad Sal Moslehian and Ghadir Sadeghi

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Abstract

We prove the generalized stability of the cubic type functional equation $$f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)$$ and another functional equation $$f(ax+y)+f(x+ay)=(a+1)(a-1)^{2}[f(x)+f(y)] +a(a+1)f(x+y),$$ where $a$ is an integer with $a \neq 0, \pm 1$ in the framework of non-Archimedean normed spaces.

Article information

Source
Real Anal. Exchange Volume 33, Number 2 (2007), 375-384.

Dates
First available in Project Euclid: 18 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.rae/1229619415

Mathematical Reviews number (MathSciNet)
MR2458254

Zentralblatt MATH identifier
05499476

Subjects
Primary: 39B22: Equations for real functions [See also 26A51, 26B25]
Secondary: 39B82: Stability, separation, extension, and related topics [See also 46A22] 46S10: Functional analysis over fields other than $R$ or $C$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05]

Keywords
generalized Hyers-Ulam stability cubic functional equation non-Archimedean space $p$-adic field

Citation

Moslehian, Mohammad Sal; Sadeghi, Ghadir. Stability of Two Types of Cubic Functional Equations in Non-Archimedean Spaces. Real Analysis Exchange 33 (2007), no. 2, 375--384. http://projecteuclid.org/euclid.rae/1229619415.


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References

  • L.M. Arriola and W.A. Beyer, Stability of the Cauchy functional equation over $p$-adic fields, Real Anal. Exchange, 31 (2005/06), 125–132.
  • S. Czerwik, Stability of Functional Equations of Ulam–Hyers–Rassias Type, Hadronic Press, Palm Harbor, 2003.
  • G.L. Forti, Hyers-Ulam stability of functional equations in several variables Aequationes Math, 50(1/2) (1995), 143–190.
  • K. Hensel, Über eine neue Begründung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math.-Verein, 6 (1897), 83–88.
  • D.H. Hyers and T.M. Rassias, Approximate homomorphisms, Aequationes Math., 44 (1992), 125–153.
  • D.H. Hyers, G. Isac and T.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.
  • K.W. Jun and H.M. Kim, The generalized Hyers–Ulam–Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 867–878.
  • K.W. Jun and H.M. Kim, On the stability of Euler–Lagrange type cubic mappings in quasi-Banach spaces, J. Math. Anal. Appl, 332 (2007), 1335–1350.
  • S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press lnc., Palm Harbor, 2001.
  • M.S. Moslehian and T.M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Disc. Math., 1 (2007), 325–334.
  • T.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62(1) (2000), 23–130.
  • T.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston and London, 2003.
  • A.M. Robert, A Course in $p$-adic Analysis, Springer–Verlag, New York, 2000.