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

TL;DR

This paper investigates the $C^1$-smooth dependence of solutions to delay differential equations on initial conditions and delay parameters, using Sobolev-type initial history spaces and analyzing translation differentiability in $L^p$.

Contribution

It establishes $C^1$-smooth dependence in Sobolev-type initial history spaces, clarifying the relationship between initial conditions, delays, and solution regularity.

Findings

Proves $C^1$-smooth dependence on initial histories and delay.

Highlights the role of translation differentiability in $L^p$ spaces.

Uses Sobolev-type spaces to handle less regular delay equations.

Abstract

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.