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Zbl 1074.34048
Jiang, Daqing; Chu, Jifeng; Zhang, Meirong
Multiplicity of positive periodic solutions to superlinear repulsive singular equations.
(English)
[J] J. Differ. Equations 211, No. 2, 282-302 (2005). ISSN 0022-0396

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 2000:
*34C25 Periodic solutions of ODE
34B16 Singular nonlinear boundary value problems
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H11 Degree theory

Keywords: periodic solutions; Hill equation; repulsive singular equations; multiplicity; superlinear

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