Robustness of nonuniform dichotomies with different growth rates

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 e for an arbitrary function ρ(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. 2010 Mathematics Subject Classification. Primary: 34D09, 34D10, 37C75.