# The Spectre of Math

## December 7, 2009

### Picard’s theorem

Filed under: Mathematics,Teaching — jlebl @ 7:59 pm

Just finished the proof of Picard’s theorem on existence and uniqueness of solutions to ODEs in my real analysis class:

Theorem (Picard). Let $I,J \subset {\mathbb{R}}$ be closed bounded intervals and let $I_0$ and $J_0$ be their interiors. Suppose $F \colon I \times J \to {\mathbb{R}}$ is continuous and Lipschitz in the second variable, that is, there exists a number L such that

$\lvert F(x,y) - F(x,z) \rvert \leq L \lvert y-z \rvert$ for all $y,z \in J$, $x \in I$.

Let $(x_0,y_0) \in I_0 \times J_0$. Then there exists an $h > 0$ and a unique differentiable $f \colon [x_0 - h, x_0 + h] \to {\mathbb{R}}$, such that

$f'(x) = F(x,f(x))$ and $f(x_0) = y_0$.

That was the last lecture for the semester, yay! The proof is essentially the standard one, but of course I don’t use the fixed point theorem on metric spaces since I don’t have that. Convergence and uniqueness is shown purely by methods taught in the class. The only thing which I had to define to state and prove the theorem was continuity in two variables. I think it’s the longest proof in the book, being 2.5 pages long (it’s 12pt font, plus it’s given in a lot of detail).

See the last section in my real analysis notes/book.

I was thinking of using something else such as implicit function theorem or some such, but Picard’s theorem is just cool. Plus the proof uses everything we’ve learned: continuity, derivatives, Riemann integral, uniform convergence, swapping of limits, etc… Plus the proof is a bunch of estimates. It doesn’t get any more “analysis” than that.