The existence of solutions homoclinic to an equilibrium for second-order time-dependent Hamiltonian systems of the type
where is a symmetric matrix, , is studied. As usual, a solution of system (1) is said to be homoclinic (to 0) if , , and as .
The existence and multiplicity of homoclinic solutions for Hamiltonian systems of this type have been studied in many recent papers via the critical point theory under the assumptions that is positive definite for all , is globally superquadratic in and the potential is periodic in .
The author studies the existence of homoclinic (to 0) solutions for a class of systems (1) when the global positive definiteness of is not necessarily satisfied and and are not periodic in . Both the case that is superquadratic in and the one that is of subquadratic growth as are considered.