## 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.

### MSC:

 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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### References:

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