New existence results on periodic solutions of non-autonomous second order Hamiltonian systems

Abstract

In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems u ̈ ( t ) + ∇ F ( t , u ( t ) ) = 0 , a.e. t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ̇ ( 0 ) − u ̇ ( T ) = 0 , where F : R × R N → R is T -periodic ( T > 0 ) in its first variable for all x ∈ R N . When potential function F ( t , x ) is either locally in t asymptotically quadratic or locally in t superquadratic, we show that the above mentioned problem possesses at least one T -periodic solutions via the minimax methods in critical point theory, specially, a new saddle point theorem which is introduced in Schechter (1998).