Volterra Equation of first kind existence of solution?

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SUMMARY

The discussion centers on the existence of solutions for the Volterra equation of the first kind, represented as u(t) = ∫_0^t K(t,τ)f(τ) dτ, where K is a compact operator from L²(0,T) to L²(0,T). The participant has established that the kernel K(t,τ) exhibits a weak singularity and that u belongs to L²(0,T). However, they are seeking references that confirm the existence of solutions for kernels that do not meet the criteria for conversion to a Volterra integral equation of the second kind, as existing literature primarily addresses cases with more favorable kernels.

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Anthony
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Hi all.

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:

  • [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.
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!

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