  In this paper, we will define the index pair $(i_A(B),\nu_A(B))$ by the dual
variational method, and show the relationship between the indices defined by
different methods. As applications, we apply the index $(i_A(B),\nu_A(B))$ to
study the existence and multiplicity of homoclinic orbits of nonlinear
Hamiltonian systems and solutions of nonlinear Dirac equations
