Abstract
We prove that the exponential dichotomy of a strongly continuous evolution family on a Banach space is equivalent to the existence and uniqueness of continuous bounded mild solutions of the corresponding inhomogeneous equation. This result addresses nonautonomous abstract Cauchy problems with unbounded coefficients. The technique used involves evolution semigroups. Some applications are given to evolution families on scales of Banach spaces arising in center manifolds theory.
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Latushkin, Y., Randolph, T. & Schnaubelt, R. Exponential Dichotomy and Mild Solutions of Nonautonomous Equations in Banach Spaces. Journal of Dynamics and Differential Equations 10, 489–510 (1998). https://doi.org/10.1023/A:1022609414870
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DOI: https://doi.org/10.1023/A:1022609414870