SMOOTHNESS OF LINEARIZATION BY MIXING PARAMETERS OF DICHOTOMY, BOUNDED GROWTH AND PERTURBATION

The paper proves that, under nonuniform dichotomy and bounded-growth hypotheses plus decay/regularity of f, the map η ↦ G(t,η)=η+w*(t;(t,η)) is a C1-diffeomorphism on an initial time interval [0, t̃j), by estimating the Jacobian ∂ηG(t,η)=I−Λ(t,η) with ∥Λ(t,η)∥<1 and invoking Plastock’s global inversion criterion (Theorem 2 with (4.8)–(4.10), and the derivative formula (4.7); see Lemma 3 for the bound on ∂ηy) . The candidate solution recovers the same structure: differentiates under the integral, derives the same type of bound for supη∥∂ηw*∥, and then proves bijectivity via the near-identity contraction argument (I+g with sup∥Dg∥<1), yielding a global C1-diffeomorphism for each fixed t in a possibly explicit interval. Small differences are cosmetic: the paper uses a sequence {θj} to widen the interval and appeals to Plastock; the model uses a single θ and a direct contraction proof and even provides an explicit threshold t̃c. Both approaches are valid and consistent with the paper’s estimates.