Semigroup-valued solutions of the Gołąb-Schinzel type functional equation (Q998232)

Let \(f\) be a function from the real line \(R\) into a semigrup \((S,0)\) and let \(g\) be a continuous mapping from \(\mathbb R\) into \(\mathbb R\). The author solves completely the functional equation \[ f(x+g(x)y) = f(x) \circ f(y) \] which generalizes the Gołąb-Schinzel equation \[ f\left( x + f(x)^k y \right) = t f(x) f(y), \] where \( t \neq 0\) is a real number and \(k\) is a positive integer. In doing this several previous results are extended.