Hi all.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

or simply [tex]u=Kf[/tex]. I've managed to prove the following:

Now I'm aware that the compactness of [tex]K[/tex] 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!

- [tex]K :L^2(0,T)\rightarrow L^2 (0,T)[/tex]
- [tex]K[/tex] is compact.
- [tex]u\in L^2(0,T)[/tex]
- The kernel [tex]K(t,\tau)[/tex] has a weak singularity.

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Volterra Equation of first kind existence of solution?

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**