Nonlinear Hamiltonian Systems With Lagrangian Boundary Conditions And Symmetrical Symplectic Capacity

This thesis consists of four chapters.In the first chapter, we give a brief introduction to the background and preliminary knowledge.In the second chapter, we first establish the so-called (L. L')-index theory for symplectic paths starting from identity associated with two Lagrangian subspaces. And we get the relationship be-tween our index (iLL′(γ),VLL′(γ)) and Galerkin approximation, saddle point reduction. Then as its applications, we consider the existence and multiplicity for asymptotically linear Hamiltonian sys-tems with arbitrary Lagrangian boundary conditions, brake solution problems and Sturm-Liouville Problems.In the third chapter, we consider a kind of abstract self-adjoint operator equations, and give some applications to the existence and multiplicity of nonlinear Hamiltonian systems with various boundary conditions.In the last chapter, we expand the definition of brake orbits to general symplectic manifolds with some symmetrical conditions, which we called symmetrical symplectic manifolds. Then we introduce the concept of symmetrical symplectic capacity for these symmetrical symplectic manifolds. By using this symmetrical symplectic capacity theory, we proved a series of existence results of symmetric closed characteristic on energy surface. One of them is that there exists at least one symmetric closed characteristic (brake orbit and S-invariant brake orbit are two examples) on prescribedψ-contact type symmetric energy surface which has a compact neighborhood with finite symmetrical symplectic capacity.