A note on the existence of multiple brake orbits

In this paper, we prove the existence of infinitely many non-constant geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for symmetric Hamiltonian systems which are superquadratic at zero and infinity by combining iteration theory of Maslov-type indices under various boundary conditions, the Galerkin approximation arguments and link theorem of critical point theory. Furthermore, we prove the existence of multiple geometrically distinct symmetric subharmonic solutions and symmetric subharmonic brake orbits for the symmetric Hamiltonian systems.