Volterra Equation of first kind existence of solution?

[Anthony]

Hi all.

I'm currently working on a problem that has led me to an integral equation of the form:

$$u(t)=\int_0^t K(t,\tau)f(\tau)\, \mathrm{d}\tau \qquad t\in (0,T)$$

or simply $$u=Kf$$. I've managed to prove the following:

• $$K :L^2(0,T)\rightarrow L^2 (0,T)$$
• $$K$$ is compact.
• $$u\in L^2(0,T)$$
• The kernel $$K(t,\tau)$$ has a weak singularity.

Now I'm aware that the compactness of $$K$$ means the inverse, if it exists, is necessarily unbounded - that's not a problem. What I can't find in any of the literature is a solid result that guarantees me existence, unless the kernel is so nice that we can convert the problem into a Volterra integral equation of the second kind: well my kernel isn't that nice. There seems to be a fair amount of stuff on these equations with weak singularities, so I'm sure it's been done!

Does anyone know of any references that deal with this stuff (journal access isn't a problem)?

 

[fresh_42]

Not sure whether it helps, but it looks interesting:
https://www.oulu.fi/sites/default/files/content/files/operator.pdf
http://math.tkk.fi/opetus/funasov/2006/WSIElectures.pdf