Differentiability with respect to delay. (English) Zbl 1021.34050

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 J. K. Hale and L. A. C. Ladeira [J. Differ. Equations 92, 14-26 (1991; Zbl 0735.34045)] which are proved for the equation \(x'(t)=f(x(t), x(t-\tau))\).


34K05 General theory of functional-differential equations
34K30 Functional-differential equations in abstract spaces
35K05 Heat equation


Zbl 0735.34045