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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:
1.
to show that there is a large class of linear differential equations admitting this general exponential behavior;
2.
to provide conditions for the existence of general dichotomies in terms of appropriate Lyapunov exponents;
3.
to establish the robustness of the exponential behavior, that is, its persistence under sufficiently small linear perturbations.

MSC:

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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