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aequationes mathematicae

, Volume 42, Issue 1, pp 271–295 | Cite as

On an alternative cauchy equation in two unknown functions. Some classes of solutions

  • Gian Luigi Forti
  • Luigi Paganoni
Research Papers

Summary

In this paper we consider the alternative Cauchy functional equationg(xy) ≠ g(x)g(y) impliesf(xy) = f(x)f(y) wheref, g are functions from a topological group (X, ·) into a group (S,·). First we prove that, ifS is a Hausdorff topological group andX satisfies some weak additional hypotheses, then (f, g) is a continuous solution if and only if eitherf org is a homomorphism. Then we describe a more general class of solutions forX =R n .

AMS (1980) subject classification

Primary 39B30, 39B50, 39B70 

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References

  1. [1]
    Aczél, J. andDhombres, J.,Functional equations containing several variables. [Encyclopedia of Mathematics and its Applications, Vol. 31]. Cambridge Univ. Press, Cambridge—New York—Melbourne, 1989.Google Scholar
  2. [2]
    Bourbaki, N.,General topology, Part 1, 2. Hermann, Paris, 1966.Google Scholar
  3. [3]
    Dhombres, J.,Some aspects of functional equations. Chulalongkorn University Press, Bangkok, 1979.Google Scholar
  4. [4]
    Dhombres, J. andGer, R.,Conditional Cauchy equations. Glasnik Mat.13(33) (1978), 39–62.Google Scholar
  5. [5]
    Forti, G. L.,La soluzione generale dell'equazione funzionale {cf(x + y) − af(x) − bf(y) − d} × {f(x + y) − f(x) − f(y)} = 0. Matematiche34 (1979), 219–242.Google Scholar
  6. [6]
    Forti, G. L.,On an alternative functional equation related to the Cauchy equation, Aequationes Math.24 (1982), 195–206.Google Scholar
  7. [7]
    Forti, G. L.,The stability of homomorphisms and amenability, with applications to functional equations. Abh. Math. Sem. Univ. Hamburg57 (1987), 215–226.Google Scholar
  8. [8]
    Forti, G. L. andPaganoni, L.,A method for solving a conditional Cauchy equation on abelian groups. Ann. Mat. Pura Appl. (4)127 (1981), 79–99.CrossRefGoogle Scholar
  9. [9]
    Forti, G. L. andPaganoni, L.,Ω-additive functions on topological groups. InConstantin Carathéodory: an international tribute, T. Rassias (Ed.). World Scientific Publ. Co., Singapore, 1990.Google Scholar
  10. [10]
    Ger, R.,On a method of solving of conditional Cauchy equations. Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 159–165.Google Scholar
  11. [11]
    Kuczma, M.,Functional equations on restricted domains. Aequationes Math.18 (1978), 1–34.CrossRefGoogle Scholar
  12. [12]
    Kuczma, M.,An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. Universytet Śląski, Warszawa-Kraków, 1985.Google Scholar
  13. [13]
    Paganoni, L.,Soluzione di una equazione funzionale su dominio ristretto. Boll. Un. Mat. Ital. (5)17-B (1980), 979–993.MathSciNetGoogle Scholar
  14. [14]
    Paganoni, L.,On an alternative Cauchy equation. Aequationes Math.29 (1985), 214–221.Google Scholar
  15. [15]
    Paganoni, L.,Remark 23. InThe Twenty-sixth International Symposium on Functional Equations April 24–May 3, 1988. Aequationes Math.37 (1989), 111.CrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag 1991

Authors and Affiliations

  • Gian Luigi Forti
    • 1
  • Luigi Paganoni
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanoItalia

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