NONLINEAR PERTURBATIONS FOR LINEAR NONAUTONOMOUS IMPULSIVE DIFFERENTIAL EQUATIONS AND NONUNIFORM (<i>H,K,µ,ν</i>)-DICHOTOMY

We explore nonlinear perturbations of a flow generated by a linear nonautonomous impulsive differential equation x′ = A(t)x, t ̸= τi,∆x|t=τi = Bix(τi), i ∈ 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, μ, ν)-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 ̸= τi,∆x|t=τi = Bix(τi) + gi(x(τi)), i ∈ Z with the help of nonuniform (h, k, μ, ν)-dichotomies.