Existence and robustness of nonuniform -dichotomies for nonautonomous impulsive differential equations

For the linear nonautonomous impulsive differential equation, we propose a new notion called the nonuniform ( h , k , μ , ν ) -dichotomy, which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity. We establish a sufficient criterion for the existence of the nonuniform ( h , k , μ , ν ) -dichotomy in terms of appropriate Lyapunov exponents. For the linear perturbation, we develop the robustness or roughness of nonuniform ( h , k , μ , ν ) -dichotomies in the sense that a nonuniform ( h , k , μ , ν ) -dichotomy persists under a sufficiently small linear perturbation.