## 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

### Keywords:

periodic solutions; fixed point index; weak singularity
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