Volterra Equation of first kind... existence of solution?

  1. 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)?
     
  2. jcsd
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