An index theory with applications to homoclinic Orbits of Hamiltonian systems and Dirac equations

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