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

In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems {<(u)double over dot>(t) + del F(t, u(t)) = 0, a.e. t is an element of [0,T], u(0) - u(T) = <(u)over dot>(0) - <(u)over dot>(T) = 0, where F : R x R-N -> R is T-periodic (T > 0) in its first variable for all x is an element of 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). (C) 2017 Elsevier Ltd. All rights reserved.