Multiplicity of positive periodic solutions to superlinear repulsive singular equations. (English) Zbl 1074.34048

The authors study the existence and multiplicity of positive periodic solutions of the perturbed Hill equation \[ x''(t) + a(t)x(t) = f(t,x(t)), \] where \(f(t,x)\) has a repulsive singularity near \(x = 0\) and is superlinear near \(x = + \infty.\) This means, respectively, that \(\lim_{x \rightarrow 0^{+}} \;f(t,x) = + \infty,\) uniformly in \(t\) and that \(\lim_{x \rightarrow + \infty} \;f(t,x)/x = + \infty,\) uniformly in \(t.\) The proof is based on a nonlinear alternative of Leray-Schauder type and Krasnoselskii’s fixed-point theorem on compression and expansion of cones.


34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
47H11 Degree theory for nonlinear operators
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