In this article, we give the existence of a smooth stable manifold which is invariant under the semiflows of the delay differential equation \( x^{\prime }= Ax(t) + Lx_t + f(t,x_t, \lambda ) \), with the assumption that the corresponding linear differential equation admits a nonuniform exponential dichotomy and the perturbation f(t, x_t, λ) is small and smooth enough. We also show that the obtained manifold is Lipschitz in the parameter λ.

中文翻译：

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无限时滞微分方程的光滑不变流形

在本文中，我们给出了一个光滑的稳定流形的存在，该流形在时滞微分方程\（x ^ {\ prime} = Ax（t）+ Lx_t + f（t，x_t，\ lambda）\ ），并假设相应的线性微分方程允许不均匀的指数二分法，并且扰动f（t，x _t，λ）小且足够平滑。我们还表明，获得的流形在参数λ中为Lipschitz 。