Theory of well-posedness for delay differential equations via prolongations and $C^1$-prolongations: its application to state-dependent delay

In this paper, we establish a theory of well-posedness for delay differential equations (DDEs) via notions of \textit{prolongations} and \textit{$C^1$-prolongations}, which are continuous and continuously differentiable extensions of histories to the right, respectively. In this sense, this paper serves as a continuation and an extension of the previous paper by this author (\cite{Nishiguchi 2017}). The results in \cite{Nishiguchi 2017} are applicable to various DDEs, however, the results in \cite{Nishiguchi 2017} cannot be applied to general class of state-dependent DDEs, and its extendability is missing. We find this missing link by introducing notions of ($C^1$-) prolongabilities, regulation of topology by ($C^1$-) prolongations, and Lipschitz conditions about ($C^1$-) prolongations, etc. One of the main result claims that the continuity of the semiflow with a parameter generated by the trivial DDEs $\dot{x} = v$ plays an important role for the well-posedness. The results are applied to general class of state-dependent DDEs.