Subharmonic Solutions For Subliner Second-order Hamiltonian Systems With Impacts

Impaction is one of the important models in nonsmooth dynamical systems. In thisarticle, we study a kind of sublinear second-order Hamiltonian systems with elastic impactionand obtain infinite subharmonic solutions.We change the impact systems into a nonsmooth dynamical systems, we obtain criticalpoints using nonsmooth saddle point theorem which corresponding to our subharmonic solu-tions. The process of the proof to our theories can be divided into three parts. The first partis obtaining the generalized gradient of varitional functions; The second part is the proof ofP.S. condition and the application of saddle point theorem; The third part is to proof theimpaction set is finite.Comparing Hamiltonian systems without impacts, we should use a technical codition.At present, this kind of conditions can't be taken away. It is a considerable problem tosimplified it or even to take it away.