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

Chu, Jifeng; Li, Ming

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69, No. 1, 276-286 (2008).

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.

Reviewer: Irena Rachůnková (Olomouc)

MSC:

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

Keywords:

positive periodic solution; Hill equation; strong singularity; weak singularity; Leray-Schauder principle; fixed point theorems in cones

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