Banach Journal of Mathematical Analysis

On the stability of the quadratic functional equation in topological spaces

M. Adam and M. Czerwik

Full-text: Open access

Abstract

In this paper we investigate the problem of the Hyers-Ulam stability of the generalized quadratic functional equation $$f(x+y)+f(x-y)=g(x)+g(y),$$ where $f,g$ are functions defined on a group with values in a linear topological space.

Article information

Source
Banach J. Math. Anal., Volume 1, Number 2 (2007), 245-251.

Dates
First available in Project Euclid: 21 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1240336223

Digital Object Identifier
doi:10.15352/bjma/1240336223

Mathematical Reviews number (MathSciNet)
MR2366108

Zentralblatt MATH identifier
1130.39021

Subjects
Primary: 39B82: Stability, separation, extension, and related topics [See also 46A22]
Secondary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.)

Keywords
quadratic functional equation Hyers-Ulam-Rassias stability in topological spaces

Citation

Adam, M.; Czerwik, M. On the stability of the quadratic functional equation in topological spaces. Banach J. Math. Anal. 1 (2007), no. 2, 245--251. doi:10.15352/bjma/1240336223. https://projecteuclid.org/euclid.bjma/1240336223


Export citation

References

  • S. Czerwik, On the Stability of the Quadratic Mapping in Normed Spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64.
  • S. Czerwik, The stability of the quadratic functional equation, In: Stability of Mappings of Hyers–Ulam type (ed. Th. M. Rassias and J. Tabor), Hadronic Press, Palm Harbor, Florida, 1994, 81–91.
  • S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, New Jersey - London - Singapore - Hong Kong, 2002.
  • S. Czerwik, Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Florida, 2003.
  • S. Czerwik, Some results on the stability of the homogeneous functions, Bull. Korean Math. Soc. 42 (2005), no. 1, 29–37.
  • S. Czerwik, On stability of the equation of homogeneous functions on topological spaces (to appear).
  • S. Czerwik and K. Dłutek, Superstability of the equation of quadratic functionals in $L^p$ spaces, Aequationes Math. 63 (2002), 210–219.
  • S. Czerwik and K. Dłutek, Quadratic difference operator in $L^p$ spaces, Aequationes Math. 67 (2004), 1–11.
  • S. Czerwik and K. Dłutek, The stability of the quadratic functional equation in Lipschitz spaces, J. Math. Anal. Appl. 293 (2004), 79–88.
  • G. L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143–190.
  • D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224.
  • D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125–153.
  • D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser Verlag, 1998.
  • S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Florida, 2000.
  • J. Rätz, On approximately additive mappings, In: General Inequalities 2 (E. F. Beckenbach, ed.), ISNM Vol. 47, Birkhäuser, Basel, 1980, 233–251.
  • S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1960.