Volume 8, Number 4, 2018, Pages 10851107 DOI：10.11948/2018.1085 
Nonlinear Perturbations for Linear Nonautonomous Impulsive Differential Equations and Nonuniform $(h,k,\mu,\nu)$dichotomy 
Jimin Zhang,Liu Yang,Meng Fan and Ming Chen 
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 GrobmanHartman 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. 
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