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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∞anfn(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 ℝN with nonempty interior.