Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.