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Archiv der Mathematik

, Volume 84, Issue 6, pp 503–511 | Cite as

On a conditional Gołab-Schinzel equation

  • Janusz Brzdek
  • Anna Mureńko
Original Paper

Abstract.

Let \(\mathbb{R}_ + : = (0,\infty ).\) We show that for every function \(f:\mathbb{R}_ + \to \mathbb{R}\) satisfying the conditional equation
$$ {\text{if }}x + f(x)y > 0,{\text{ then }}f(x + f(x)y) = f(x)f(y) $$
either there exists a solution \(g:\mathbb{R} \to \mathbb{R}\) of the Gołab-Schinzel equation
$$ g(x + g(x)y) = g(x)g(y) $$
such that \(f = g|_{\mathbb{R}_ + } \) (i.e., f(x) = g(x) for \(x \in \mathbb{R}_ + \)) or there is x0 > 0 with f(x0) < −1 and f(x) = 0 for x ≠ x0 . In particular we determine the solutions \(f:\mathbb{R}_ + \to \mathbb{R}\) of the conditional equation that are continuous at a point, Lebesgue measurable or Baire measurable (i.e., have the Baire property). In this way we solve some problems raised by the first author.

Mathematics Subject Classification (2000).

39B22 

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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.Department of MathematicsPedagogical UniversityKrakówPoland
  2. 2.Department of MathematicsUniversity of RzeszówRzeszówPoland

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