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Nonlinear perturbations for linear nonautonomous impulsive differential equations and nonuniform $(h,k,\mu,\nu)$-dichotomy
Keywords:Nonautonomous impulsive differential equations, topological equivalence, nonuniform $(h,k,\mu,\nu)$-dichotomy, invariant manifolds.
Abstract:
      We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation $x''=A(t)x,t\neq\tau_i,\Delta x|_{t=\tau_i}=B_ix(\tau_i),~ i \in \Z$ in Banach spaces. Here we assume that the linear nonautonomous impulsive equation admits a more general dichotomy on $\R$ called the nonuniform $(h,k,\mu,\nu)$-dichotomy, which extends the existing uniform or nonuniform dichotomies and is related to the theory of nonuniform hyperbolicity. Under nonlinear perturbations, we establish a new version of the Grobman-Hartman theorem and construct stable and unstable invariant manifolds for nonlinear nonautonomous impulsive differential equations $x''=A(t)x+f(t,x)$, $t\neq \tau_i, \Delta x|_{t=\tau_i}=B_ix(\tau_i)+g_i(x(\tau_i)),i \in \Z$ with the help of nonuniform $(h,k,\mu,\nu)$-dichotomies.