Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel.
I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where
- $v:\mathbb R \to \mathbb R$ is the unknown,
- $A, B, C, D$ are four given real constants.
I am interested in existence, uniqueness (and maybe - let's dream - exact formulas for solutions).
I think the problem is not trivial, so I look first for periodic solutions $v$. If any solution exists, then we have the following equations for the Fourier coefficients:
$$ \forall n \in \mathbb{Z} \quad 0 = A~\delta_0(n) + B~\hat{\varphi}(n)~c_n(v) + C~c_n(v) + D~c_n(v~(\varphi*v))$$
So if I could find a link (an equation) between $c_n(v)$ and $c_n(v~(\varphi*v))$, I could be almost done ! But is it possible ?