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:
to show that there is a large class of linear differential equations admitting this general exponential behavior;
to provide conditions for the existence of general dichotomies in terms of appropriate Lyapunov exponents;
to establish the robustness of the exponential behavior, that is, its persistence under sufficiently small linear perturbations.


34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
34D10 Perturbations of ordinary differential equations
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