C1-smooth dependence on initial conditions and delay: Spaces of initial histories of sobolev type, and differentiability of translation in Lp

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Abstract

The objective of this paper is to clarify the relationship between the C1-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 C1-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 C1-smooth dependence on initial histories and delay holds by adopting spaces of initial histories of Sobolev type, where the differentiability of translation in Lp plays an important role.

Original languageEnglish
Article number91
JournalElectronic Journal of Qualitative Theory of Differential Equations
Volume2019
DOIs
Publication statusPublished - 2019

Keywords

  • Constant discrete delay
  • Delay differential equations
  • Differentiability of translation in L
  • History spaces of Sobolev type
  • Smooth dependence on delay

ASJC Scopus subject areas

  • Applied Mathematics

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