Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. (English) Zbl 1148.34025

The paper proves existence and multiplicity of positive periodic solutions of the perturbation of Hill’s equation \[ x''+a(t)x=f(t,x)+e(t), (1) \] 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.


34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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