I am trying to solve the equation:

$\phi(x)=\int_{-\infty}^\infty K(x, t)\phi(t)dt$

for $K$ given $\phi$. This closely resembles a Fredholm Integral of the Second Kind, which has the form:

$\phi(x)=f(x)+\lambda \int_{-\infty}^\infty K(x, t)\phi(t)dt$

with $\phi$ unknown and $\lambda$, $K$, $f$ known. Is Fredholm Theory helpful, even though I'm solving for a different term in the expression than usual?