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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##.
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##.
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