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