A quick guide to ordinary state-dependent delay differential equations

TL;DR

This paper reviews the well-posedness of ordinary differential equations with state-dependent delays, emphasizing fixed point methods and Sobolev spaces to establish existence, uniqueness, and continuous dependence of solutions.

Contribution

It provides a streamlined approach to prove well-posedness for state-dependent delay differential equations using fixed point theory and exponential Sobolev spaces.

Findings

Existence and uniqueness of solutions for small times

A blow-up criterion for global solutions

Continuous dependence on initial data and right-hand sides

Abstract

We review $H^{1}$-well-posedness for initial value problems of ordinary differential equations with state-dependent right-hand side. We streamline known approaches to infer existence and uniqueness of solutions for small times given a Lipschitz-continuous prehistory. The paramount feature is a reduction of the differential equation to a fixed point problem that admits a unique solution appealing to the contraction mapping principle. The use of exponentially weighted Sobolev spaces in this endeavor proves to be as powerful as for ordinary differential equations without delay. Our result includes a blow-up criterium for global existence of solutions. The discussion of well-posedness is concluded by new results covering continuous dependence on initial prehistories and on the right-hand sides.