Multiple Periodic Solutions For Second Order Hamiltonian Equations With Oscillatory Potential

In this paper,we deal with the existence and multiplicity of periodic solutions for second order Hamilton Differential equations x" + g(x)= p(t)with oscillatory potentials or weakly oscillatory potentials,by using Poincare-Birkhoff theorem and topology degree on successor mappings.If we take the equation x"+g(x)= p(t)as a perturbation of the autonomous equation x" + g(x)= 0 under the force p(t),the oscillatory potential condition can be regarded as a non-degenerate condition at infinity on the existence of infinitely many periodic solutions.Tongren Ding,Iannacci and Zanolin firstly applied Poincare-Birkhoff twist theorem to second order Hamiltonian differential equations with oscillatory potentials,and then through the analysis of properties for time map Dingbian Qian improved their work without global Lipschitz condition and semilinear condition.Later,Capietto,Mawhin and Zanolin,Zaihong Wang,Dunyuan Hao and Shiwang Ma also extended the previous works.Among them,Dunyuan Hao and Shiwang Ma considered weak oscillatory potential problems in semilinear condition.Recently,Fonda and Zaihong Wang considered issues related to singular equations.The purpose of this paper is to consider the problem of singular equations with oscil-latory potentials and weak oscillatory potentials under the condition of further results.We will consider the equation with oscillatory potential in a relatively easy checked integral condition and consider the equation with weakly oscillatory potential in non-semilinear condition.The results generalize the works mentioned above.It is not easy to make esti-mations of the phase-plane analysis similar to the previous article for our weak conditions.So this paper does not consider the twist property of Poincare map,but considering the twist property of the successor map.This benefit is only to consider variations of angles of some special solutions.In order to obtain the required estimates,we also analyze the time map of the autonomous equation in detail.Then we apply Poincare-Birkhoff twist theo-rem to prove the existence of infinitely many subharmonic solutions of the corresponding problems.Due to the fact that the Massera theorem is not established in the case of the singular equation,we prove the existence of the periodic solution of the corresponding problem by using the topological degree theory.