The author considers the iterated equation (1) \(G(f(x), f^{n_1} (x), \cdots, f^{n_k} (x)) = F(x)\), where \(f^0(x) = x\), \(f^k(x) = f \cdot f^{k - 1} (x)\), \(k = 1, 2, \cdots\). Define the set \(A = \{\varphi \in C^0(I,I) : \varphi (a) = a\), \(\varphi(b) = b\), there is \(M > 0\) such that for all \(x_1, x_2 \in I = [a,b]\) and \(x_1 > x_2\), \(0 \leq \varphi(x_1) - \varphi(x_2) \leq M(x_1 - x_2)\}\). The author gives some sufficient conditions for the existence, uniqueness and stability of the solution of (1) on the set \(A\).