Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond Peano's theorem: a variational look at discontinuous ODE systems

Pablo Pedregal

Abstract

We propose a framework to define solutions of ODE systems under a novel condition that goes well beyond the usual continuity condition required in the classical theory of ODEs (Peano's or Picard's theorems). We illustrate our results with some simple but enlightening examples, including some facts about Sobolev fields, and mention some relevant questions to proceed with this analysis further.

Beyond Peano's theorem: a variational look at discontinuous ODE systems

Abstract

We propose a framework to define solutions of ODE systems under a novel condition that goes well beyond the usual continuity condition required in the classical theory of ODEs (Peano's or Picard's theorems). We illustrate our results with some simple but enlightening examples, including some facts about Sobolev fields, and mention some relevant questions to proceed with this analysis further.

Paper Structure

This paper contains 6 sections, 7 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Proof of main result
  3. A legitimate extension
  4. Some illustrative examples
  5. The case of Sobolev fields
  6. Final comments

Key Result

Theorem 1.1

Let $\mathbf{f}(\mathbf{z}):\mathbb{R}^N\to\mathbb{R}^N$ be self-continuous, and for every $\mathbf{z}\in\mathbb{R}^N$. Then the initial-value, Cauchy problem admits at least one generalized, absolutely continuous solution $\mathbf{x}(t)$ for every $\mathbf{x}_0\in\mathbb{R}^N$, and every positive $T$.

Theorems & Definitions (14)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 3.1
  • ...and 4 more