MAT 531 (Spring 1996) Topology/Geometry II

Week 10
This week we covered material in Spivak, Chapter 5. The main theorem is the following (Spivak, Theorem 5.2).

Theorem 1. Suppose f: U --> R^n is a function defined on an open set U in R^n. Suppose x_0 is a point of U and that the closed ball B = B_{2a}(x_0) is contained in U, for some a > 0. Suppose that f satisfies a Lipschitz condition on B, i.e. that there exists a number K such that

|f(x) - f(y)| < K |x - y|

for every x,y in B. Such a function is automatically continuous on B. Let L be an upper bound for |f| on B, and choose b > 0 such that b < = a/L and that b < 1/K.

Then for each x in the closed ball B_a(x_0) there exists a unique solution alpha_x : (-b,b) --> U to the initial value problem

alpha'(t) = f(alpha(t)),
alpha(0) = x.

This theorem is proved using the contraction lemma (similarly to its use in the proof of the inverse function theorem).

We also proved the following (Spivak, Theorem 5.4).

Theorem 2. With the same notation, the map

alpha : (-b,b) x B_a(x_0) --> U

defined by alpha(t,x) = alpha_x(t) is continuous in both variables.

And we agreed to accept without proof the theorem (see Lang1 or Lang2) that if f is of class C^k, then so is alpha.

(Back to Week-by-week page)