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Collisionless orbits of singular and non singular dynamical systems. (English) Zbl 1120.34029

The authors study the existence of \(T\)-periodic solutions for a nonlinear system of nonautonomous differential equations of the type \[ \begin{aligned} \pm u''+k^{2}u=f(t,u), \end{aligned} \] where \(k\in\mathbb{R}\) and \(f(t,u)\) is a continuous vector valued function \(T\)-periodic in \(t\), allowed to have a singularity at \(u=0\). Apparently, for the first time in the literature neither positivity nor a constant sign behaviour of the nonlinearity is supposed. The proofs are based on fixed point theory for completely continuous operators including a new type of cone. The obtained results improve recent work even in the scalar case.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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