In this paper, we study two composite functional equations \(\begin{aligned} f(x) = x f(1 - x + f(x)) \end{aligned}\) and \(\begin{aligned} f(x + f(y)) = (x + f(y))f(1 - x + f(x)), \end{aligned}\) where \(f : \mathbb {R} \rightarrow \mathbb {R}\) . These functional equations are associated with the Dirichlet function. We study many general properties of these functional equations. In addition, we study monotonic solutions of these equations.
