By means of Schauder fixed point theorem and Banach contraction principle, we investigate the existence and uniqueness of Lipschitz solutions of the equation 𝒫 ( f ) ∘ f = F . Moreover, we get that the solution f depends continuously on F . As a corollary, we investigate the existence and uniqueness of Lipschitz solutions of the series-like iterative equation ∑ n = 1 ∞ a n f n ( x ) = F ( x ) ,     x ∈ 𝔹 with a general boundary restriction, where F : 𝔹 → 𝔸 is a given Lipschitz function, and 𝔹 , 𝔸 are compact convex subsets of ℝ N with nonempty interior.