Monotonic solutions of a composite functional equation

Chao first considered the composite functional equation $$\begin{aligned} f(x + 2f(y)) = f(x) + y + f(y) \quad (x, y \in \mathbb {R}), \end{aligned}$$ where $$f: \mathbb {R} \rightarrow \mathbb {R}$$ . This equation was studied by several mathematicians. In this paper, we study the monotonic solutions of the composite functional equation $$\begin{aligned} f(xyf(y)) = f(x) y f(y) \quad (x, y \in \mathbb {R}), \end{aligned}$$ where $$f: \mathbb {R} \rightarrow \mathbb {R}$$ . This equation is similar to Chao’s equation.