1
$\begingroup$

Is it possible to use the implicit function theorem to prove the existence of ordinary differential equations?

I have seen a proof for the existence of ordinary differential equations in which the conditions are that:

We consider an initial value problem where a function and it's partial with respect to y are continuous.

However these are two of the conditions for the implicit function to work. So I feel like there should be a way to connect the two.

$\endgroup$
  • 1
    $\begingroup$ In Lang's Real Analysis the infinite dimensional version of the inverse function theorem is used to prove existence and, in particular, smooth dependance on the initial values for ODEs, by applying it to the associated operator acting on function spaces. But this does not seem to be what you have in mind, is it? $\endgroup$ – Thomas May 7 '15 at 20:30
  • $\begingroup$ Not quite what I was looking for. But still helpful! $\endgroup$ – Devon White May 7 '15 at 20:57
1
$\begingroup$

It is possible to use the implicit function theorem to prove uniqueness of solutions of equations of certain type. The connection is more transparent for systems of the form $$ \frac{dx}{dt}=f(x,y),\qquad \frac{dy}{dt}=g(x,y) \tag{1}$$ which include the standard ODE $y'=g(t,y)$ as a special case (let $f\equiv 1$, so that $x\equiv t$).

Suppose there is a function $\psi$ of $x,y$ such that $\nabla \psi = (-g,f)$. Then (1) implies $$\frac{d}{dt}\psi(x(t),y(t)) = 0$$ and therefore every trajectory of (1) is contained in a level set of $\psi$. If the Implicit Function theorem applies to $\psi$, i.e., if it's $C^1$ smooth with nonzero gradient, then the level sets are smooth curves. In particular, they do not branch, and this implies uniqueness for the solution of (1).

The existence of $\psi$ (locally) is equivalent to the equality $-g_y=f_x$, which simply says that $(f,g)$ is a field of zero divergence. So, (1) has a unique solution (for given initial values) under this condition.

One can generalize by observing that it suffices to have $\nabla \psi = (-g,f) h$ with some scalar function $h$.

Closely related: Bendixson–Dulac theorem.

$\endgroup$
  • $\begingroup$ Yes, can you elaborate a little bit more for me? I'm confused about how the Implicit Function Theorem implies that the level sets are smooth curves. And, I am wondering how the local existence of $$\psi$$ is equivalent to the equality $$-g_y =f_x$$ $\endgroup$ – Devon White May 11 '15 at 4:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.