The Solutions Of Hamiltonian Systems With Nonlinear Boundary Value Conditions

Let X be a real infinite-dimensional separable Hilbert space with norm || · || and inner product(·,·).Let A:D(A)(?)X?X be an unbounded self-adjoint and invertible operator satisfying ?(A)= ?d(A).Assume that Y is a Banach space with the norm || · ||y satisfying D(A)(?)Y(?)X,the inclusion map from D(A)to Y is compact and the inclusion from Y to X is continuous.Assume N:Y?X is continuous,M:Y?Y is compact and satisfies ||M(x)||Y<p for all x?Y and some p>0.Consider the operator equation x = A-1Nx + Mx.(0.0.1)By using of topological degree methods and the index theory established for lin-ear operator equations,we obtain some results for the existence of solutions and non-trivial solutions of(0.0.1)assuming that Nx satisfies asymptotically linear conditions.The results can be applied to Hamiltonian systems with different nonlinear boundary value conditions and yield some existence theorems for solutions under asymptoti-cally linear conditions.We investigate second order Hamiltonian systems with impulsive effects satisfy-ing nonlinear Picard boundary value conditions and generalize Lees' existence result to those systems.We establish an index theory for linear Hamiltonian systems satisfying gener-alized periodic boundary value conditions with coefficient matrices in L1 and as ap-plications we investigate asymptotically linear Hamiltonian systems via critical point theory.We generalize some existence results for solutions of second order Hamiltonian systems with periodic potentials to operator equations.