Existence of solution to "weird" integral equation

Question

in my current work I come across an integral of the form \begin{align} x(a) = \int_{\Omega} f(a, u) x(u) du \end{align} where $\Omega$ is $\Omega \subseteq \mathbb{R}$, e.g. $\Omega = (0,1)$. The functions $f$ and $x$ are differentiable and integrable on $\Omega$. I am interested in conditions for the existence of a solution $x(t)$ to the above integral. My problem is that the integral does not fit into the usual form on integral equations, i.e. something along the lines of \begin{align} x(a) = \int_{-\infty}^a f(k, u) x(u) du. \end{align} Do you know of any works regarding the existence of a solution to the first integral equation? A formula for explicit solutions would of course also be nice. Thank you!