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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L. B. and C. V. were supported by FCT/Portugal through UID/MAT/04459/2013
D. D. was supported in part by an Australian Research Council Discovery Project DP150100017, Croatian Science Foundation under the project IP-2014-09-2285 and by the University of Rijeka research grant 13.14.1.2.02.
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Barreira, L., Dragičević, D. & Valls, C. Admissibility on the half line for evolution families. JAMA 132, 157–176 (2017). https://doi.org/10.1007/s11854-017-0017-4
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DOI: https://doi.org/10.1007/s11854-017-0017-4