$C^1$ -对初始条件和时滞的光滑依赖：Sobolev型初始历史空间和$L^p$中平移的可微性
$C^1$-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in $L^p$
摘要：
本文的目的是阐明时滞微分方程（DDE）解对初始历史的$C^1$ -光滑依赖性（即，初始条件）和延迟参数。为此，我们考虑一类包含常数离散时滞的DDE。从微分方程理论的观点来看，$C^1$ -光滑依赖性问题是基本的.然而，由于相应的泛函微分方程相对于延迟参数具有较小的规律性，因此上述关系并不明显。本文采用Sobolev型初始历史空间，证明了$C^1$ -光滑对初始历史和时滞的依赖性成立，其中平移的可微性在$L^p$中起着重要作用.
英文摘要：
The objective of this paper is to clarify the relationship between the $C^1$-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class of DDEs which include a constant discrete delay. The problem of $C^1$-smooth dependence is fundamental from the viewpoint of the theory of differential equations. However, the above mentioned relationship is not obvious because the corresponding functional differential equations have the less regularity with respect to the delay parameter. In this paper, we prove that the $C^1$-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in $L^p$ plays an important role.
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