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In this paper, we establish a theory of well-posedness for delay differentialequations (DDEs) via notions of \textit{prolongations} and\textit{$C^1$-prolongations}, which are continuous and continuouslydifferentiable extensions of histories to the right, respectively. In thissense, this paper serves as a continuation and an extension of the previouspaper by this author (\cite{Nishiguchi 2017}). The results in \cite{Nishiguchi2017} are applicable to various DDEs, however, the results in \cite{Nishiguchi2017} cannot be applied to general class of state-dependent DDEs, and itsextendability 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 mainresult claims that the continuity of the semiflow with a parameter generated bythe trivial DDEs $\dot{x} = v$ plays an important role for the well-posedness.The results are applied to general class of state-dependent DDEs.

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