I'm considering a nonlinear integral equation $F:U \subset \mathcal{L}^2(a,b) \to \mathcal{L}^2(a,b)$, where $a,b \in \mathbb{R}$ and $H$ is a constant. The operator is defined as
$$ F[x](t)=\int_a^b \ln{ \left( \frac{(t-s)^2+H^2}{(t-s)^2+(x(s)-H)^2} \right) } ds.$$
Furthermore I'm considering the restriction $H-x(t) <0 $, for all $t \in [a,b].$
I would like to show if the problem is ill-posed or not and why.
The problem is taken from TIKHONOV AND ARSENIN Solution of Ill-Posed Problems (p.13-15).
By now I found out that it is a special Fredholm integral operator called Uryson operator. And my idea is to show that the operator is a compact operator, thus there does not exist a continuous inverse of $F$, and contradict the properties for a well-posed problem by Hadamard.
I would be greatful for ideas and inspiration how to show the uniqueness and the compactness.