

A function f(x, y) continuous on a rectangle R satisfies a Lipschitz condition with constant
L if

Theorem 1.1. (Picard’s existence theorem):
 y' = f(x, y) with y(a) = b has a solution in the rectangle R := {(x, y) : x − a ≤ h, y − b ≤ k} provided:
 P(i): (a) f is continuous in R, with bound M (so f(x, y) ≤ M) and (b) Mh ≤ k.
 P(ii): f satisfies a Lipschitz condition in R.
 Furthermore, this solution is unique.

what is Gronwall's inequality


what is P(iii) (condition for a global soln.)

what is Picard's existence theorem

