Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $C^1$-smooth dependence on initial conditions and delay: spaces of
  initial histories of Sobolev type, and differentiability of translation in
  $L^p$   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.

19
The categories of this paper are: eess
13
The categories of this paper are: quant-ph
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th