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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.

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up vote 3 down vote accepted

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.

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