Differentiability with respect to delay (Q1978218)

The authors deal with the initial value problem \(x'(t)=f(t,x(t),x(t-\tau)), t>0, x(t)=\phi (t), -r \leq t \leq 0,\) where \(f: \Omega \rightarrow {\mathbb{R}}^n\) and \(\Omega \) is open in \({\mathbb{R}}_+ \times {\mathbb{R}}^{2n}\). The differentiability with respect to the delay to the solution of this problem is proved. Further, an infinite-dimensional version of the problem is investigated and results on the existence, uniqueness, continuous dependence and differentiability with respect to \(\tau \) are obtained. An application to heat conduction is also given. The authors extend the earlier results by \textit{J. K. Hale} and \textit{L. A. C. Ladeira} [J. Differ. Equations 92, 14-26 (1991; Zbl 0735.34045)] which are proved for the equation \(x'(t)=f(x(t), x(t-\tau))\).