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.
For example:
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.
Then
,
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. 🙂