The Existence Of Solutions For Non-autonomous Second-order Hamiltonian Systems

This paper mainly studies two kinds of non-autonomous second order Hamiltonian sys-tems.Firstly,we are concerned with a class of non-autonomous second order Hamiltonian systems:???where T>0,B?t? is an N×N symmetric matrix,continuous and T-periodic in.Potential function F?t,x?:[0,T]×R^N?R is superquadratic for.By using saddle point theorem,we get the existence of one periodic solution.Particularly,if potential function is even,we get the multiplicity results of the periodic solutions with the help of fountain theorem.Secondly,we consider the following perturbed order Hamiltonian systems:???where ??1 is a parameter,L?t? is positive definite for all t?R and f?L²(R,R^N)\{0}small enough.By introducing a local superquadratic condition,we get the existence of two nontrivial homoclinic solutions.Our result is obtained by the mountain pass theorem and Ekeland's variational principle.