The Existence and Uniqueness Theorem of Ordinary Differential Equations (statement)
I have to take a little detour for now to prove an important result: the existence and uniqueness theorem of ordinary differential equations. This is one of those hard analytic nubs that differential geometry takes as a building block, but it still needs to be proven once before we can get back away from this analysis.
Anyway, we consider a continuously differentiable function defined on an open region , and the initial value problem:
for some fixed initial value . I say that there is a unique solution to this problem, in the sense that there is some interval and a unique function satisfying both conditions.
In fact, more is true: the solution varies continuously with the starting point. That is, there is an interval around , some neighborhood of and a continuously differentiable function called the “flow” of the system defined by the differential equation , which satisfies the two conditions:
Then for any we can get a curve defined by . The two conditions on the flow then tell us that is a solution of the initial value problem with initial value .
This will take us a short while, but then we can put it behind us and get back to differential geometry. Incidentally, the approach I will use generally follows that of Hirsch and Smale.