ISL 1992 Problem 2

Let $\mathbb{R}^+$ be the set of all non-negative real numbers. Given two positive real numbers $a$ and $b,$ suppose that a mapping $f: \mathbb{R}^+ \mapsto \mathbb{R}^+$ satisfies the functional equation:

$$f(f(x)) + af(x) = b(a + b)x.$$

Prove that there exists a unique solution of this equation.