aequationes mathematicae

, Volume 59, Issue 3, pp 248–254 | Cite as

Bounded solutions of the Golab—Schinzel equation

  • J. Brzdek


Let \( {\Bbb K} \) be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \( {\Bbb K} \) n be a positive integer, and \( f : X \to {\Bbb K} \) be a solution of the functional equation¶¶\( f(x + f(x)^n y) = f(x) f(y) \).¶We prove that, if there is a real positive a such that the set \( \{ x \in X : |f(x)| \in (0, a)\} \) contains a subset of second category and with the Baire property, then f is continuous or \( \{ x \in X : |f(x)| \in (0, a)\} \) for every \( x \in X \). As a consequence of this we obtain the following fact: Every Baire measurable solution \( f : X \to {\Bbb K} \) of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set \( A \subset X \) with \( f(X \backslash A) = \{ 0 \}) \).


Positive Integer Functional Equation Complex Number Bounded Solution Equal Zero 
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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • J. Brzdek
    • 1
  1. 1.Department of Mathematics, Pedagogical University, Rejtana 16A, P-35-310 Rzeszów, Poland PL

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