Multiple solutions of singular periodic problems

In this contribution we consider the singular periodic problem $$u^{\mathrm{\prime }\mathrm{\prime }}+ku=g(u)+e(t),u(0)=u(2\pi ),u^{\mathrm{\prime }}(0)=u^{\mathrm{\prime }}(2\pi ),$$ where $k\in \mathbb{R},$ $e:\text{J}\to \mathbb{R}$ is Lebesgue integrable and $g:\text{I}\mapsto \mathbb{R}$ is continuous, but can have a singularity at $x=0.$ In particular, I wish to revisit our earlier result regarding the existence of at least two its solutions and present a proof that is somewhat more transparent than the original one.