Suppose we have a homogeneous Volterra equation of the second kind: $$ x(t) = \int_0^t K(t,s,x(s))\,\mathrm{d}s, \qquad 0\leq t\leq T, $$ and the sign of $K$ is determined by only $x$ for all $t$ and $s$. A natural conclusion for me is to think is that if a nontrivial solution exists, it must be nonnegative or nonpositive. But for whatever reason, I can't find a direct proof of this claim, which makes me question its validity. For my purposes, I do not need to approximate $x$ and I don't care what it is, just that it has one sign.
Thank you.