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.