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.

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.