A neat fact regarding Cauchy problems for infinite systems

[wrobel]

Hi everyone,

I would like to share a result regarding the existence of solutions for systems of ordinary differential equations with an arbitrary number of variables.

Let $\Gamma$ be an arbitrary nonempty set of indices. For each $\gamma \in \Gamma$, let $f_\gamma$ be a scalar-valued function depending on a finite number of variables such that:

$$f_\gamma \in C([t_1, t_2] \times \mathbb{R}^{n_\gamma}) \quad \text{and} \quad \sup_{[t_1, t_2] \times \mathbb{R}^{n_\gamma}} |f_\gamma| \le M_\gamma$$
Theorem. The Cauchy problem

$$\dot{x}_\gamma = f_\gamma(t, x_{\sigma_1}, \dots, x_{\sigma_{n_\gamma}}), \quad x_\gamma(t_1) = \hat{x}_\gamma, \quad \gamma \in \Gamma$$
where $\{\sigma_1, \dots, \sigma_{n_\gamma}\} \subset \Gamma$ is a finite subset of indices dependent on $\gamma$, has a solution $\{x_\gamma(t)\}_{\gamma \in \Gamma}$ such that $x_\gamma \in C^1[t_1, t_2]$ for each $\gamma \in \Gamma$.

 

[wrobel]

I am confident that this theorem is not new. Its proof relies heavily on Zorn's lemma, which is also not surprising

 

[JesseProbst]

Thanks for sharing this result. Dealing with an arbitrary number of variables in Cauchy problems is always tricky. Does this existence theorem also imply uniqueness under these conditions, or are there additional constraints required for that?

 

[wrobel]

Uniqueness does not hold under these conditions, even in the finite-dimensional case.