In the book "ODE and Dynamical systems" (2012) by Gerald Teschl, Thm 4.1 p.111, one finds the complex version of the local Cauchy-Lipschitz theorem which the author explains, is proved in the same way as the real case (but he points out a difference when one looks for a maximal solution).
One naturally wishes to have the global version but I cannot find it anywhere and I've learned to be wary of such situation as the statement is usually wrong. The global real version claims that for an ODE of the form $$ \left\lbrace \begin{aligned} \mathbf{X}'(t) & = F\big(t,\mathbf{X}(t)\big)\\ \mathbf{X}(t_0) & = \mathbf{X}_0 \end{aligned} \right. $$ with $F: \Omega \subset I\times E \to E $ continuous ($I$ interval, $E$ Banach space) and (globally) Lispchitz continuous in the second variable, there exists a unique solution of the initial value problem defined on the whole of $I$. cf. for example wikipedia French or wikipedia German.
- Of course, in the complex case, $F$ should be holomorphic and one should replace $I$ with a simply connected open subset of $\mathbb{C}$ and $E$ with $\mathbb{C}^n$.
- As I'm writing the question (I added a domain $\Omega$), I see that one should not define $F$ on an unbounded region in the "second" variable as a holomorphic function will go to infinity, as well as its derivative, excluding the possibility for $F$ to be Lipschitz continuous.
So assuming in addition that $\Omega$ is bounded, can something else go wrong?
