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Nonlinear Perturbations for Linear Nonautonomous Impulsive Di?erential Equations and Nonuniform (h,k,u,v)-dichotomy
Keywords:Nonautonomous impulsive di?erential equations; Nonuniform (h,k,u,v)-dichotomy; Topological equivalence; Invariant manifolds
      We explore nonlinear perturbations of a °ow generated by a linear nonautonomous impulsive di?erential equation x0 = A(t)x; t 6= ?i;¢xjt=?i = Bix(?i); i 2 Z in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general di- chotomy on R called the nonuniform (h; k; 1; o)-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlin- ear perturbations, we establish a new version of the Grobman-Hartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive di?erential equa- tions x0 = A(t)x + f(t; x); t 6= ?i;¢xjt=?i = Bix(?i) + gi(x(?i)); i 2 Z with the help of nonuniform (h; k; 1; o)-dichotomies.