Nontrivial Solutions Of Hamiltonian Systems With Lagrangian Boundary Conditions And Brake Subharmonic Solutions Of Hamiltonian Systems

In this paper, we consider two problems of the Hamiltonian systems: the problem of the existence of nontrivial solutions and the property of the L (or (L', L "))-Maslov type index for the Hamiltonian systems with Lagrangian boundary conditions and the problem of the brake subharmonic solutions for the first order non-autonomous Hamiltonian systems.This paper consists of three parts.The first part, i.e., chapter 2, we give some preliminaries, mainly recall the Maslov-type index theory for symplectic paths associated with Lagrangian subspaces and an iteration theory for the L₀-Maslov type index theory. All the details can be found in [35,37,41,68].The second part, i.e., chapter 3 and 4, we use the L-Maslov type index theory , the Galerkin approximation method and the variational methods to study the existence of nontrivial solutions and the property of the L-Maslov type index for the Hamiltonian systems with Lagrangian boundary conditionswhere J = (?) is the standard symplectic matrix, I_n is the unit matrix oforder n,T＞0,H∈C²([0,T]×R²ⁿ,R),(?)H(t,z) is the gradient of H(t, z) respect toz,and L is a Lagrangian subspace in symplectic vector space (R²ⁿ,ω₀),ω₀=(?) dy_i. This problem has relation with the Arnold chord conjecture (see [4]), and the brake orbit problem in the autonomous case, and has relation with the Lagrangian intersection problem when generalizing the symplectic space R²ⁿ to a general symplectic manifold M and L is a Lagrangian subspace of M.In chapter 3, the Hamiltonian function H is given bywhere (?) is a semipositive symmetrical continuous matrix for all t∈[0, T] and (?) satisfies a superquadratic condition at infinity. We get Theorem 3.1.1 about the existence of nontrivial solutions of the problem (0.1) and the property of the L-Maslov type index. In addition, we study the general casewhere L' and L" are any two linear Lagrangian subspaces of E²ⁿ. For the problem (0.2), we obtain Theorem 3.1.2 about the existence result and the property of the (L' ,L")-index. In this chapter, we give detailed proof of Theorem 3.1.1 and3.1.2.In chapter 4, we use the same methods as in chapter 3 to study the existence of nontrivial solutions of the problem (0.1) for which are not convex, unbounded, sub-quadratic and not uniformly coercive, and obtain a property of the L-Maslov index of this nontrivial solution of (0.1). We get Theorem 4.1.1 about these results and give a detailed proof of it.The third part, i.e., chapter 5, we mainly use the Galerkin approximation method and the iteration inequalities of the L₀-Maslov type index theory in [41] to study the problem of the brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that for any j∈N, there exists a jT-periodic nonconstant brake solution z_j such that z_j and z_kj are distinct (k≥5), see Theorem 5.1.1 and 5.1.2. Up to the author's knowledge, Theorem 5.1.1 and 5.1.2 are the first results for the brake subharmonic solution problem for the time being.