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.

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