Iteration Inequalities of the Maslov-Type Index Theory with Applications

In this paper, by using the ω-index theory introduced by Y. Long in (1999, Pacific J. Math.187, 113–149), in particular, the splitting numbers, Maslov-type mean index, and the homotopy component of symplectic matrix, we establish various inequalities of the Maslov-type index theory for iterations of symplectic paths starting from the identity. As an application, these results are used to study Rabinowitzʹ conjecture on the prescribed minimal period solution problem of nonlinear Hamiltonian systems