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The objective of this paper is to clarify the relationship between the$C^1$-smooth dependence of solutions to delay differential equations (DDEs) oninitial histories (i.e., initial conditions) and delay parameters. For thispurpose, 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 thetheory of differential equations. However, the above mentioned relationship isnot obvious because the corresponding functional differential equations havethe less regularity with respect to the delay parameter. In this paper, weprove that the $C^1$-smooth dependence on initial histories and delay holds byadopting spaces of initial histories of Sobolev type, where thedifferentiability of translation in $L^p$ plays an important role.

$C^1$-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in $L^p$