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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.

34C25Periodic solutions of ODE
34B16Singular nonlinear boundary value problems for ODE
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H11Degree theory (nonlinear operators)
Full Text: DOI
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