We propose a new approach toward the existence and uniqueness of periodic solutions to linear and semilinear evolution equations. Our approach is based on the connection of the conditional stability of evolution families (i.e., stability only in a subspace of the Banach space containing the initial data) with the choice of the initial data from which emanates the periodic solution. We also give applications to exponentially dichotomic evolution families as well as to nonautonomous damped wave equations.

中文翻译：

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演化方程解的条件稳定性和周期性

我们针对线性和半线性发展方程的周期解的存在性和唯一性提出了一种新方法。我们的方法是基于进化族的条件稳定性（即仅在包含初始数据的Banach空间的子空间中的稳定性）与产生周期解的初始数据之间的联系。我们还将其应用于指数二分式演化族以及非自治阻尼波方程。