In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo, São Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.

在本文中，我们研究了无限维Banach空间中整体小扰动下非自治线性常微分方程的指数二分法的鲁棒性。对于具有任意小的周期的快速振荡扰动的情况，已经获得了一些应用，这表明即使在这种情况下，稳定性也很强。这些结果扩展到了无限的维度，这在Coppel（稳定性理论中的二分法。数学讲义，Springer，柏林，1970年）中给出。设在罗德里格斯岛（Invariânciapara sistemas deequaçõesdifferenciais com delayamento eaplicações，Tese de Mestrado，圣保罗大学，圣卡洛斯，1970）和科洛登和罗德里格斯岛（非线性肛门74：2695–2719等，2011）。（无穷维非自治线性微分方程的稳定性问题。arXiv：1906.04642，2019）我们使用称为广义概周期函数的函数类别来扩展概周期函数的通常类别，并适合于对这些振荡扰动进行建模。我们还提供了先前结果的无穷大示例。