Differentiable solutions of an equation with product of iterates

The paper proves existence and uniqueness of differentiable solutions to the product-of-iterates equation on R+ by conjugating to a polynomial-like equation on R, defining L_f(x)=∑_{k=1}^n λ_k f^{k-1}(x), and applying a Banach contraction to T(f)=L_f^{-1}∘F with careful estimates for the constants K0,…,K6 and M' = M*/(λ1−K0 M^2) under λ1>K0 M^2 and K<1. The candidate solution reproduces the same strategy in logarithmic variables (φ = log g∘exp, H = log G∘exp), with F_φ mirroring L_f and the fixed-point map S~=F_φ^{-1}∘H. The contraction bounds and the choice M' = M*/(λ1−K0 M^2) agree with the paper’s Theorem 3.1 and its proof, including the two-part Lipschitz estimates analogous to (3.4)–(3.5). Minor differences are notational (e.g., K3 vs. K'_3) and organizational (working first on I then extending), but the mathematical content and logic are essentially the same as the paper’s proof. See the theorem statement and proof details and the contraction estimates in Theorem 3.1 and Lemma 2.12 in the PDF , the reduction via conjugation and class definitions , and the contraction inequalities (3.4)–(3.5) .