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 $F:U\to\mathbb{R}^n$ defined on an open region $U\subseteq\mathbb{R}^n$ , and the initial value problem:

$\displaystyle\begin{aligned}v'(t)&=F(v(t))\\v(0)&=a\end{aligned}$

for some fixed initial value $a\in U$ . I say that there is a unique solution to this problem, in the sense that there is some interval $(-a,a)$ and a unique function $v:(-a,a)\to\mathbb{R}^n$ satisfying both conditions.

In fact, more is true: the solution varies continuously with the starting point. That is, there is an interval $I$ around $0\in\mathbb{R}$ , some neighborhood $W$ of $a$ and a continuously differentiable function $\psi:I\times W\to U$ called the “flow” of the system defined by the differential equation $v'=F(v)$ , which satisfies the two conditions:

$\displaystyle\begin{aligned}\frac{\partial}{\partial t}\psi(t,u)&=F(\psi(t,u))\\\psi(0,u)&=u\end{aligned}$

Then for any $w\in W$ we can get a curve $v_w:I\to U$ defined by $v_w(t)=\psi(t,w)$ . The two conditions on the flow then tell us that $v_w$ is a solution of the initial value problem with initial value $w$ .

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.

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May 4, 2011 - Posted by John Armstrong | Analysis, Differential Equations

4 Comments »

[...] we can start actually closing in on a solution to our initial value problem. Recall the [...]

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[...] that we’ve got the existence and uniqueness of our solutions down, we have one more of our promised results: the smooth dependence of solutions on initial conditions. That is, if we use our existence and [...]

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[...] should look familiar, since they’re very similar to the conditions we wrote down for the flow of a differential [...]

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