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Zbl 1148.34025
Chu, Jifeng; Li, Ming
Positive periodic solutions of Hill's equations with singular nonlinear perturbations. (English)
[J] Nonlinear Anal., Theory Methods Appl. 69, No. 1, A, 276-286 (2008). ISSN 0362-546X

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.
[Irena Rach\uu nková (Olomouc)]

MSC 2000:: *34B30 Special ODE
34B18 Positive solutions of nonlinear boundary value problems
34C25 Periodic solutions of ODE
34B16 Singular nonlinear boundary value problems

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

Cited in: Zbl 1170.34325

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