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A quick guide to ordinary state-dependent delay differential equations

Bernhard Aigner, Marcus Waurick

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.

A quick guide to ordinary state-dependent delay differential equations

Abstract

We review -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.

Paper Structure

This paper contains 11 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The delay
  4. The time derivative and its inverse
  5. The evaluation operator
  6. Main results
  7. Applications
  8. Constant delay
  9. An academic example
  10. Integro-differential equations
  11. Conclusion

Key Result

Theorem 1.1

Let $-\infty<a<b<\infty$. Then $H^{1}(a,b;H)\hookrightarrow \mathcal{C}([a,b];H)$. More precisely, every $f\in H^{1}(a,b;H)$ admits an absolutely continuous representative and

Theorems & Definitions (31)

  • Theorem 1.1: Sobolev-embedding
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2: Continuity with respect to time
  • Proposition 2.3
  • proof
  • Remark 2.4: range of $I_{\rho}$ and time derivative
  • Theorem 2.5
  • proof
  • ...and 21 more