A Landesman–Lazer-type condition for asymptotically linear second-order equations with a singularity

We consider the $T$-periodic problem $$x''+g(t, x)=0,$$ $$x(0)=x(T), \quad x'(0)=x'(T),$ where $g:[0, T] \times ]0, +\infty[ \to \mathbb{R}$ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large $x$, $g(t, x)$ is controlled from both sides by two consecutive asymptotes of the $T$-periodic Fu\v{c}ik spectrum, with possible equality on one side. Using a suitable Landesman-Lazer-type condition, we prove the existence of a solution.