Periodic Solutions For A Class Of Nonautonomous Second Oredr Hamilton Systems

The paper is concerned with periodic solutions to nonautonomous second order Hamilton systemsWhere, M : [0,T] - S(Rn,Rn) is a continuous mapping in the space S(Rn,Rn) of symmetric real(n x n)- matrices, such that for some u > 0 and all(t,z) [0,T] x Rn, (M(t)x,x) > u|x|2.A S(Rn,Rn),F : [0,T] x Rn R is continuous and F :[0,T]xR R exists,is continuous andWe study the existence of periodic solutions of the systems by using Ekeland variational principle and the saddle points theorem.We suppose that the nonlinearity VF and potential F belongs to a class of unbounded functional.Our work improves the existed results.We obtained the results of multiplicity of periodic solutions of the systems by using Lusternik-Schnirelman Category theory and the generalized saddle points theorem ,and the functional does not need the condition of constant definite .At last,we obtained the existence of infinity many distinct periodic solutions of the corresponding non-perturbation systems by using the Symmetric mountain pass theorem.