Homogeneous Fredholm equation of the second kind

Hi,

during the analysis of a problem in my phd thesis

I have resulted in the following equation.

$\displaystyle \varphi(x)= \int_a^b K(x,t)\varphi(t)dt$

which is clearly a *homogeneous Fredholm equation of the second kind*

The problem is that I can't find in any text any way of solving it.

Solutions are provided only for special cases like when the kernel K

is symmetric

$\displaystyle K(x,t)=K(x,t)$

or when it is separable which are both not my case.

The particular form of the equation I am dealing with is

$\displaystyle \varphi(x)= \int_a^b \Lambda(x,t)g(x)\varphi(t)dt$

where $\displaystyle \Lambda(x,t)$ is symmetric and depends only in the difference x-t, $\displaystyle \Lambda(x,t)=\Lambda(x-t)$ , and g(x) a known function involving logarithm.

Any ideas of how to deal with this kind of form?

Thank you in advance