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.
|Journal||Electronic Journal of Qualitative Theory of Differential Equations|
|Publication status||Published - 2019|
- Constant discrete delay
- Delay differential equations
- Differentiability of translation in L
- History spaces of Sobolev type
- Smooth dependence on delay