The general Volterra Equation of the second kind in convolution form can be described by:

$$ \phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a $$

Suppose we wish to prove a bounded solution $\phi$ exists. There are many results for when $a\leq x \leq b$, but what about $x\in[0,\infty)$?

In particular, I'm looking for conditions on $K(x-t)$ for existence of such a solution. I do not care what the solution is, just to prove it exists.

Can anyone please point me in the right direction?

Thank you.

Edit: I am aware that the contraction mapping theorem provides a sufficient condition. I am looking for a weaker condition since my $K$ is not necessarily nonnegative, and therefore,

$$ \int_a^\infty |K(x)| \mathrm{d}x \leq C $$ is a very bad bound.

for every bounded$f$ or something else? $\endgroup$ – fedja Jul 16 '17 at 0:50I just wanted to be sure there were no overarching theorems to cover my problem.-- You never know that much (after all, what you are going to create yourself is exactly a theorem of that type), but even if there are such theorems, I don't think that should bother you in the slightest. Just use the tools you have in any way you can and try to keep the arguments reasonably clear and elegant. After all, mathematics, as I see it, is about being able to create what you want with what is at your disposal, not about searching old manuscripts for universal recipes. Otherwise, you are welcome :-) $\endgroup$ – fedja Jul 16 '17 at 16:33