1992 IMO 第 2 题

(CHN 1) Let $\mathbb{R}^{+}$be the set of all nonnegative real numbers. Given two positive real numbers $a$ and $b$, suppose that a mapping $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ satisfies the functional equation $$f(f(x))+a f(x)=b(a+b) x$$ Prove that there exists a unique solution of this equation.

(CHN 1) 令 $\mathbb{R}^{+}$ 为所有非负实数的集合。给定两个正实数 $a$ 和 $b$，假设映射 $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ 满足函数方程 $$f(f(x))+a f(x)=b(a+b) x$$ 证明该方程存在唯一解。