Existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance

We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.