This week we shall prove theorem 3.10 on page 86, the Picard Lindelöf existence and uniqueness theorem. Every thing is in the book on pages 72 - 87. I'll only do the first proof.
We start with background stuff:
I'll the def of the Picard iteration, then recall the def of the Lipschitz condition.
Speak about the sup-norm anduniform convergence and the function spaces C(J,R^n). A little about metric spaces
Cauchy sequences, convergence and completeness.
Then the Fix point theorem for contracting maps (theorem 3.4).
Then, finally, we'll prove th 3.10