The Existence Of Periodic Solutions For Some Class Of Hamiltonian Systems

In this paper,we begin with the non-autonomous second-order Hamiltonian systems where B(t)=(bij(t))∈C(0,T;RN×N)is a symmetric matrix withT/2-periodic in t.For the superquadratic case,by using the fountain theorem in critical point theory,we prove that the system(S1)has infinitely many periodic solutions.For the subquadratic case,we obtain the existence of infinitely many solutions for system(S1)by using the mini-max method.In both cases,our solutions consist of three infinite solution sequences,two of which are sequences of odd functions and the other is a sequence of even functions.Next,we consider the following autonomous second-order Hamiltonian systems where V(x)denotes the potential function.Under the subquadratic condition and the even type condition,we prove that problem(S2)has at least five periodic solutions.Especially,two solutions possess the minimal period.The proof is based on the least action principle and the minimax methods in critical point theory.Finally,we study a class of impulsive differential systems where x=(x1,x2,…,xN),F∈C1([0,T]×RN,R),▽F denotes the gradient of F in x,Iij∈C(RN,R)denote the impulses occurring at the instants tjwith j∈Z\{0},0<t1<…<tl<T and tj+l=tj.Using the least action principle,we give the existence of three solutions with minimal period for system(S3).