Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries

In this paper, we define a relative Morse index for two continuous symmetric matrices paths in R2n satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries.