A Green Function Approach to Smooth Nonautonomous Topological Equivalence with Unbounded Nonlinearities under $(\mu,\nu)$--Dichotomies

Abstract

We study the smooth topological equivalence between a nonautonomous linear system on the positive half-line and a quasilinear perturbation whose nonlinear part is not assumed to be globally bounded with respect to the state variable. The linear equation is assumed to admit a $(\mu,\nu)$--dichotomy, and the construction is carried out through the Green operator associated with this dichotomy. The usual global boundedness of the perturbation is replaced by locally uniform Green-integrability conditions along the relevant linear and nonlinear solutions. Under a smallness condition involving the global Lipschitz constant of the perturbation and the Green operator, we first construct Palmer-type maps which give a continuous topological equivalence on $\mathbb R^+$. We then impose Green-integrability conditions on the successive variational equations and a further first-order smallness condition which guarantees the invertibility of the derivative of the inverse map. Under these assumptions, the equivalence is of class $C^r$. We also provide a class of examples for which the perturbation is unbounded with respect to the space variable and all the hypotheses can be verified directly.