The Study Of Multiple Periodic Solutions For A Class Of Damped Vibration Systems

In this paper,we mainly study the multiplicity of periodic solutions for the following damped vibration system:whereu(u=u(t)?C²(R,R^N),I_N×Nis theN×N identity matrix,q(t)?L¹(R;R)is T-periodic and satisfies?₀^Tq(t)dt?0,A(t)?[a_ij(t)]is a T-periodic symmetricN×N matrix valued function with a_ij?L^?(R;R),(i,j?,1,2,…,N),B=[b _ij]is an antisymmetricN×N constant matrix,F_u(t,u)denotes its gradient with respect to the u variable,and F satisfies the superquadratic condition at infinity:Under some suitable conditions,the existence of infinitely many periodic solutions of the above system can be obtained by using variational methods and the fountain theorem of Bartsch.The details are as follows:In Chapter 1,we briefly describes the research background,research status and research content of this paper.In Chapter 2,we introduce the necessary preparatory knowledge and main results,and give the corresponding examples to illustrate our results,and also give the improvement and expansion of the results compared with related results.In Chapter 3,we give the variational setting and the fountain theorem of Bartsch.By variational methods,we transform the problem of periodic solutions for damped vibration systems into the problem of critical points for the corresponding functional.In Chapter 4,we prove the Cerami condition is satisfied,which will be needed in the proofs of the two theorems in this paper.In Chapter 5,the multiplicity of periodic solutions of damped vibration systems is proved by using the fountain theorem of Bartsch.In Chapter 6,we give the summary and prospect of this paper.