Real Analysis Exchange

Stability of the Cauchy functional equation over p-adic fields.

L. M. Arriola and W. A. Beyer

Full-text: Open access

Abstract

This paper corrects some errors found in [2], which discusses an extension of Lorentz transformations over a non-Archimedean valued field; namely, the $p$-adic field $\mathbb{Q}_{p}$. The paper [2] is based on the results given by Hyers [7] which showed that for a continuous function $f$ defined on $\mathbb{R}$, the Cauchy functional equation $f(x + y) = f(x) + f(y)$ is stable. By stable we mean that if there exists $\epsilon >0$ such that $\| f(x + y) - f(x) - f(y) \| < \epsilon$, $\forall x,y$, then there exists a unique and continuous $\mathcal{L}$ such that $\| \mathcal{L}(x) - f(x) \| \leq \epsilon$, $\forall x$ and $\mathcal{L}(x + y) = \mathcal{L}(x) + \mathcal{L}(y)$. In this paper, we show this result is true on the $p$-adic field $\mathbb{Q}_p$.

Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 125-132.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516796

Mathematical Reviews number (MathSciNet)
MR2218193

Zentralblatt MATH identifier
1099.39019

Subjects
Primary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 39B22: Equations for real functions [See also 26A51, 26B25]

Keywords
Cauchy functional equation p-adic field stability

Citation

Arriola, L. M.; Beyer, W. A. Stability of the Cauchy functional equation over p -adic fields. Real Anal. Exchange 31 (2005), no. 1, 125--132. https://projecteuclid.org/euclid.rae/1149516796


Export citation

References

  • G. Bachman, Introduction to $p$-adic Numbers and Valuation Theory, Academic Press, (1964).
  • W. A. Beyer, Approximately Lorentz Transformations and $p$-adic Fields, LA–DC–9486, Los Alamos National Laboratory, (1968).
  • C. J. Everett and S. M. Ulam, On Some Possibilities of Generalizing the Lorentz Group in the Special Relativity Theory, J. of Comb. Theory, 1 (1966), 248–270.
  • G. Forti, Hyers-Ulam Stability of Functional Equations in Several Variables, Aequationes Mathematicae, 50 (1995), 143–190.
  • F. Q. Gouvêa, $p$-adic Numbers, Springer-Verlag, (1997).
  • K. Hensel, Theorie der Algeraischen Zahlen, Tuebner, Leipzig and Berlin, (1908).
  • D. H. Hyers, On the stability of the linear functional equation, Proc. N.A.S., 27 (1941), 222–224.
  • D. H. Hyers and T. Rassias, Approximate Homomorphisms, Aequationes Mathematicae, 44 (1992), 125–153.
  • A. Khrennikov, $p$-adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers, (1994).
  • A. Khrennikov, Non–Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, (1997).
  • L. Székelyhidi, Ulam's Problem, Hyers Solution-and to Where they led, Stability of Functional Equations, Kluwer Academic Publishers, (2000), 259–285.
  • S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, Inc., (1960) (2000), 63–69.