Positive periodic solutions of Hill's equations with singular nonlinear perturbations (Q929655)

The paper proves existence and multiplicity of positive periodic solutions of the perturbation of Hill's equation \[ x''+a(t)x=f(t,x)+e(t), \leqno(1) \] where \(a(t),e(t)\) are continuous, \(T\)-periodic functions. The nonlinearity \(f(t,x)\) is continuous in \((t,x)\) and \(T\)-periodic in \(t\) and has a singularity at \(x=0\). The case of a strong singularity as well as that of a weak singularity is considered, and \(e\) does not need to be positive. The proofs are based on Krasnoselskii's fixed point theorem in cones and on the Leray-Schauder alternative together with a truncation technique. Some recent results in the literature are generalized and improved.