Admissibility on the half line for evolution families |
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Authors: | Luis Barreira Davor Dragičević Claudia Valls |
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Institution: | 1.Departamento de Matemática,Instituto Superior Técnico, Universidade de Lisboa,Lisboa,Portugal;2.School of Mathematics and Statistics,University of New South Wales,Sydney,Australia |
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Abstract: | For an arbitrary evolution family, we consider the notion of an exponential dichotomy with respect to a family of norms and 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. |
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