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Functions having the Darboux property and satisfying some functional equation

Eliza Jabłońska — 2009

Colloquium Mathematicae

Let X be a real linear topological space. We characterize solutions f:X → ℝ and M:ℝ → ℝ of the equation f(x+M(f(x))y) = f(x)f(y) under the assumption that f and M have the Darboux property.

Baire measurable solutions of a generalized Gołąb–Schinzel equation

Eliza Jabłońska — 2010

Commentationes Mathematicae

J. Brzdęk [1] characterized Baire measurable solutions $f : X \to K$ of the functional equation $f (x + f {(x)}^{n} y) = f (x) f (y)$ under the assumption that $X$ is a Fréchet space over the field $K$ of real or complex numbers and $n$ is a positive integer. We prove that his result holds even if $X$ is a linear topological space over $K$ ; i.e. completeness and metrizability are not necessary.

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