On a conditional Gołab-Schinzel equation

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.