Minimum Periodic Solution Of A Second-order Even Hamiltonian Syste

The minimal periodic problem of Hamiltonian system originated in 1978 of Rabinowitz,he proposed a conjecture that first-and second-order Hamiltonian systems have nonconstant periodic solutions with any prescribed minimal period.Ekeland and Hofer in 1985 made an important progress and confirmed that the conjecture is true for first-order Hamiltonian system with strictly convex assumptions.Long in 2021 made a remark on Rabinowitz’s conjecture,stated that the conjecture still remains open,and proposed many open problems.In this thesis,we mainly study the minimal periodic solutions of the second-order even Hamiltonian systems without any convexity assumptions.The thesis is organized as follows.In Chapter one,we mainly introduces the background of the periodic solutions of Hamiltonian systems and the research situation at home and abroad.In Chapter two,we mainly study the minimal periodic solutions of second-order even Hamiltonian systems under the Ambrosetti-Rabinowitz condition(superquadratic condition).In this chapter,we develop a new method(combining discrete critical point theory and approximation)to study the minimal periodic solutions of second-order even Hamiltonian systems without any convexity assumptions.Firstly,we discretize the second-order continuous Hamiltonian systems according to different step lengths,and apply the discrete critical point theory to study the nonnegative solutions of the Dirichlet boundary value problems of the corresponding second-order discrete Hamiltonian system.Then,based on the obtained nonnegative solutions,we construct a sequence of continuous functions which can be shown to be precompact.The limit function of the convergent subsequence is the solution of the second-order continuous Hamiltonian systems.Finally,considering the symmetry of the potential,we obtain the minimal periodic solutions of the second-order continuous Hamiltonian systems.In Chapter three,we mainly studies the minimal period problem of second-order Hamiltonian systems with asymptotically linear conditions.In this chapter,we overcome the difficulties arising from the absence of Ambrosetti-Rabinowitz condition,and apply the method proposed in the previous chapter(combining discrete critical point theory and approximation)to study the existence of periodic solutions with prescribed minimal period for second-order even Hamiltonian systems with asymptotically linear nonlinearities.