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.π

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