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.


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