Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given \begin{document}$ T>0 $\end{document}, there exists a central symmetric periodic solution with minimal period \begin{document}$ T $\end{document}. Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either \begin{document}$ T $\end{document} or \begin{document}$ T/3 $\end{document}. Also we give some other lower bound estimations for brake orbits case.