We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that $(\phi, T)$ is a solution of the Cauchy problem
$$\left\{\begin{array}{l} y'(t) = F(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$
and deduce the following theorem:
Theorem 2: If $F$ is a continuous function, then there exists at least one solution of the Cauchy problem (1).