SMOOTH LINEARIZATION OF NONAUTONOMOUS COUPLED SYSTEMS

The paper proves C1-regularity of the conjugacies Hn and H̄n with respect to each variable by (i) an explicit Green-operator series for h̄n (equation (31)) derived from the coupled/uncoupled systems (5)–(6) and the Green operator G defined in (7), and (ii) dominated term-by-term differentiation relying on hypotheses (H1)–(H13) and bounds via Ck,n, Dk,n, Mk,n; see Theorem 2.1 for existence of conjugacies, the representation (31), and the derivative bounds (14) and (33). From these, Theorem 3.3 (ξ ↦ H̄n is C1), Theorem 3.4 (ξ ↦ Hn is C1 under (H7)), Theorem 3.16 (η ↦ H̄n is C1), and Theorem 3.17 (η ↦ Hn is C1 under (H7)) follow. The candidate solution matches the representation and domination strategy, but differs in technique by using explicit variational equations for Dx2 and the inverse function theorem to conclude C1 regularity of Hn once Dξ h̄n is bounded by Kn+Jn+|G(n,n+1)|γn<1. These are compatible with the paper’s results and assumptions (H1)–(H13) and reproduce the same conclusions, albeit with a slightly different route (paper differentiates the identity H̄n∘Hn=Id to obtain Dhn, whereas the candidate invokes the Banach-space inverse function theorem). No substantive logical gaps or missing hypotheses were found in either argument. Citations: definition of G and hypotheses (H1)–(H13) , existence of conjugacies (Theorem 2.1) , representation of h̄n (equation (31)) and η-regularity proof (Theorem 3.16) , ξ-regularity proof including (14) and Theorem 3.4 , and η-regularity bounds (33) and (H13) ; the definitions/bounds for Dk,n and Mk,n appear in the lemmas preceding (33) ; the hypotheses list including (H6)–(H13) is summarized in one place .