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 ) , ν 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 ) , ν A ( B ) ) to study the existence and multiplicity of homoclinic orbits of nonlinear Hamiltonian systems and solutions of nonlinear Dirac equations.