First, we prove a lemma for contraction map :
It is easy to prove that when is small or has small norm, then this map is a contraction map. However, if we know that is a kernel such that
then our conclusion still holds.
Consider Fourier transformation and non-linear ODE.
with sufficient good properties for .
Then use :
The correspondent functional is:
It seems we have done most work, but still the convolution product is annoying. We should eliminate it in a technical way.
Restrict out solution in some ball of space.
when is sufficient small, is a contraction map.
The rest work is to verify that well-defined.
However, if we encounter a similar problem:
, where , and satisfies some good properties on the boundary.
We can still use Fourier Transformation to solve this. 🙂