Solution of a differential matrix equation Given a differential matrix equation, ie $X'=A(z)X+B(z)$ where both $A$ and $B$ are matrix of size $n\times n$ with coefficients that are holomorfic functions in a convex open set $\Omega$ and continuous on the closure $\bar \Omega$,  and an initial data: $X(z_0)=u$, I know there exists a solution.
However, I haven't been able to find on the internet a proof of the existence. So the question is how to prove it.
I already know it when $A(z)$ has constant coefficients, but it cannot be extended to this case.
Also I've read about Magnus Series. Although I don't fully understand them, I'd prefer a easier proof of the existence, as I'm not really interested in a generic formula.

Picard iteration produces a sequence of approximations
that converges uniformly to a solution: 
Let $X_0(z)=u$ and find $X_n(z)$ ($n\ge1$) to satisfy
$$ X_n(z) = u+ \int_{z_0}^z (A(w)X_{n-1}(w)+B(w))\,dw
$$
integrating along the line segment path connecting $z_0$ to $z$.
Each $X_n$ is holomorphic on $\Omega$ since $\Omega$, being  convex,
is simply connected.
From the given assumptions, 
it is straightforward to prove by induction that
$$
\|X_{n}(z)-X_{n-1}(z)\| \le \frac{C^n|z-z_0|^n}{n!} \qquad (n\ge1),
$$
(in terms of a matrix norm) where
$C=\sup_{z\in\Omega}( \|A(z)\|(|u|+1)+\|B(z)\|)<\infty$. 
Uniform convergence to a solution follows 
since the telescoping series
$ \sum_{n=1}^\infty (X_n(z)-X_{n-1}(z) )$
is absolutely convergent.
