I have to prove the Picard-Lindelof theorem:

Let $I\subset \mathbb{R}$ and $U\subset \mathbb{R}^n$ be open sets and let $f:I\times U\rightarrow \mathbb{R}^n$ be continuous and Lipshitz continuous by $y$ and $(x_0,y_0)\in I\times U$; then the Cauchy problem: $$y'(x)=f(x,y(x)),\; \; y(x_0)=y_0$$ has a unique local solution if $f$ is differentiable.

I have found a lot of proofs that use terms from Banach spaces and etc. Can someone give me a sketch of how to prove this without using Banach spaces and fixed points?

  • $\begingroup$ Look up, e.g., in the book by Hartman, ODE. $\endgroup$ – Artem Feb 29 '16 at 19:59

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