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The paper deals with the boundary value problem \[ u^{\prime \prime }+k\,u=g(u)+e(t),\quad u(0)=u(2\pi ),\,\,u^{\prime }(0)=u^{\prime }(2\pi ), \] where $k\in \mathbb{R}$, $g\:I\mapsto \mathbb{R}$ is continuous, $e\in \mathbb{L}J$ and $\lim _{x\rightarrow 0+}\int _x^1g(s)\,\hspace{0.56905pt}\text{d}s=\infty .$ In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

Resonance and multiplicity in periodic boundary value problems with singularity