engleski

Admissibility on the half line for evolution families

For an arbitrary evolution family, we consider the notion of an exponential dichotomy with respect to a family of norms and we characterize it completely in terms of the admissibility of bounded solutions, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms we recover the notion of a nonuniform exponential dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of an exponential dichotomy with respect to a family of norms under sufficiently small Lipschitz and $C^1$ parameterized perturbations. Moreover, we establish the optimal regularity of the dependence on the parameter of the projections onto the stable spaces of the perturbation.

Exponential dichotomies ; robustness

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