Aequationes mathematicae

This paper is concerned with the functional equation f (p (z) + bf (z)) = h (z) for f. By constructing a convergent power series solution of an auxiliary equation of the form $$ g\left(\beta ^{2}z\right) -p\left(g(\beta z)\right) =bh\left( g(z)\right), $$ analytic solutions of the form $$ f(z) = \left\{ g\left( \beta g^{-1}(z)\right) -p(z)\right\} /b $$ for the original equation are obtained.

Let A be an additive semigroup of real numbers the additive group generated by which is non-cyclic. Let $$I=(a,b)$$ be an open interval and $$\mathcal {A}=\left\{ f^\alpha :\alpha \in A\right\} $$ be an iteration semigroup of fixed point free increasing functions of I onto I such that $$\alpha <\beta $$ implies $$f^\alpha