Existence and multiplicity of periodic solutions for some second order Hamiltonian systems

In this paper, we study the existence of nontrivial periodic solutions for the second order Hamiltonian systems (t) + del F(t,u(t)) = 0, where F(t, x) is either nonquadratic or superquadratic as vertical bar u vertical bar ->infinity Furthermore, if F(t, x) is even in x, we prove the existence of infinitely many periodic solutions for the general Han-tiltonian systems (t) + A(t)u(t) + V F (t, u (t)) = 0, where AO is a continuous T-periodic symmetric matrix. Our theorems mainly improve the recent result of Tang and Jiang [X.H. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl. 59 (2010) 3646-3655].