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Robustness of nonuniform dichotomies with different growth rates. (English) Zbl 1272.34067
Summary: For nonautonomous linear differential equations $v'=A(t)v$ in a Banach space, we consider general exponential dichotomies that extend the notion of (uniform) exponential dichotomy in various ways. Namely, the new notion allows: stable and unstable behavior with respect to growth rates $\varepsilon^{c\rho(t)}$ for an arbitrary function $\rho(t)$; nonuniform exponential behavior, causing that any stability or conditional stability may be nonuniform; and different growth rates in the uniform and nonuniform parts of the dichotomy. Our objective is threefold: {\parindent=6mm\item{1.} to show that there is a large class of linear differential equations admitting this general exponential behavior; \item{2.} to provide conditions for the existence of general dichotomies in terms of appropriate Lyapunov exponents; \item{3.} to establish the robustness of the exponential behavior, that is, its persistence under sufficiently small linear perturbations.\par}

34D09Dichotomy, trichotomy
34G10Linear ODE in abstract spaces
34D10Stability perturbations of ODE