Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.