Existence theorems Volterra Equation of second kind on unbounded domains

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.

• Do you want the existence of a bounded solution for every bounded $f$ or something else? – fedja Jul 16 '17 at 0:50
• Great point. Assume f is something like e^(-x). – AD500712838 Jul 16 '17 at 2:03
• The only general idea I can come up with is to honestly go to the Fourier side and try to see if $\widehat f/(1-\widehat K)$ has a bounded inverse Fourier transform (so $1-\widehat K$ should have no zeroes in the upper half-plane and everything should behave nicely in general). This is, of course, just a restatement of the question. On the other hand, if you have some particular kernel and particular function (class of functions) in mind, it may be much more productive to just tell us what they are. – fedja Jul 16 '17 at 12:17
• In a research problem, I came across a class of kernels that are useful for a different reason. However, I'm trying to show that those same kernels produce solutions to this equation. Fedja: I think you pretty much answered my question. If I am understanding your answer, you are saying this problem definitely depends on the kernels themselves, given $f(x) = e^{-x}$. I am aware of techniques to answer my question for specific kernels. I just wanted to be sure there were no overarching theorems to cover my problem. Thank you! – AD500712838 Jul 16 '17 at 15:25
• --I 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 :-) – fedja Jul 16 '17 at 16:33