On a counterexample in connection with the Picard-Lindel\"of theorem

TL;DR

This paper presents a counterexample showing that the Lipschitz condition in the Picard-Lindel"of theorem cannot be replaced by Lipschitz continuity in $x$, and discusses numerical methods for finding multiple solutions.

Contribution

It provides a specific counterexample illustrating limitations of the Lipschitz condition in the theorem and explores numerical approaches to identify multiple solutions.

Findings

The classical Euler method detects only one solution in the example.

The counterexample demonstrates the necessity of the Lipschitz condition.

Numerical adjustments can find multiple solutions.

Abstract

We give an example, which demonstrates that in the situation of the Picard-Lindel\"of theorem, the Lipschitz condition on the right hand side $f(x,y)$ with respect to $y$, cannot be replaced by Lipschitz continuity in $y$ for every $x$. We show that, in our example, the classical Euler method detects only one of infinitely many solutions and we outline how the latter can be adjusted to find also other solutions numerically.