On second-order differentiability with respect to parameters for differential equations with state-dependent delays

TL;DR

This paper establishes first and second-order differentiability of solutions to differential equations with state-dependent delays concerning parameters, under certain monotonicity assumptions, enhancing understanding of parameter sensitivity.

Contribution

It provides new results on the second-order differentiability of solutions with respect to parameters for a class of delay differential equations, assuming piecewise strict monotonicity.

Findings

First and second-order differentiability in a pointwise sense.

Differentiability in the C-norm on the state-space.

Applicability to equations with piecewise strictly monotone delays.

Abstract

In this paper we consider a class of differential equations with state-dependent delays. We show first and second-order differentiability of the solution with respect to parameters in a pointwise sense and also using the C-norm on the state-space, assuming that the state-dependent time lag function is piecewise strictly monotone.