I've got the following problem I'm working on which is related to some of my research:

Solve:

$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$

for f, given $G$ which has whatever smoothness properties you might need.

There are two possible simplifications that can be made regarding G: either

1: $G(x,y) \equiv 0$ when $y>x$ and thus we continue the integral up to $\infty$

or

2: $G$ is symmetric, $G(x,y) = G(y,x)$ (but the integral must extend to x, not $\infty$)

These are the only bounds I have so far to put on G.

Hoping you guys might have some ideas. This question comes about in a few problems related to the study of the structure of solids.

-edit

I'm not really looking for a general solution, since its highly unlikely it will be possible. I'm really interested in finding particular solutions, assumptions, or techniques that might be helpful in understanding the problem.

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