Nontrivial solutions for asymptotically linear hamiltonian systems with Lagrangian boundary conditions

2012 ◽  
Vol 32 (4) ◽  
pp. 1545-1558
Author(s):  
Liu Chungen ◽  
Zhang Qingye
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Nontrivial Solutions ◽  
Asymptotically Linear ◽  
Linear Hamiltonian Systems ◽  
Lagrangian Boundary

scholarly journals The Oscillation Numbers and the Abramov Method of Spectral Counting for Linear Hamiltonian Systems

2021 ◽  
Vol 248 ◽  
pp. 01002
Author(s):  
Julia Elyseeva
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Spectral Counting ◽  
Hamiltonian Problems ◽  
Differential Systems ◽  
Linear Hamiltonian Systems ◽  
Strict Normality

In this paper we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. For the Hamiltonian problems we do not assume any controllability and strict normality assumptions which guarantee that the classical eigenvalues of the problems are isolated. We also omit the Legendre condition for their Hamiltonians. We show that the Abramov method of spectral counting can be modified for the more general case of finite eigenvalues of the Hamiltonian problems and then the constructive ideas of the Abramov method can be used for stable calculations of the oscillation numbers and finite eigenvalues of the Hamiltonian problems.

Periodic Solutions of Asymptotically Linear Hamiltonian Systems without Twist Conditions

2010 ◽  
Vol 65 (5) ◽  
pp. 445-452
Author(s):  
Rong Cheng ◽  
Dongfeng Zhang
Keyword(s):  
Dynamical System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Galerkin Approximation ◽  
Point Theory ◽  
Dynamical System Theory ◽  
Hamiltonian Form ◽  
Asymptotically Linear ◽  
Linear Hamiltonian Systems ◽  
Existence Of Periodic Solutions

In dynamical system theory, especially in many fields of applications from mechanics, Hamiltonian systems play an important role, since many related equations in mechanics can be written in an Hamiltonian form. In this paper, we study the existence of periodic solutions for a class of Hamiltonian systems. By applying the Galerkin approximation method together with a result of critical point theory, we establish the existence of periodic solutions of asymptotically linear Hamiltonian systems without twist conditions. Twist conditions play crucial roles in the study of periodic solutions for asymptotically linear Hamiltonian systems. The lack of twist conditions brings some difficulty to the study. To the authors’ knowledge, very little is known about the case, where twist conditions do not hold.

scholarly journals J-Self-Adjoint Extensions for a Class of Discrete Linear Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Guojing Ren ◽  
Huaqing Sun
Keyword(s):  
Boundary Conditions ◽  
Hamiltonian Systems ◽  
Limit Point ◽  
Limit Circle ◽  
Linear Hamiltonian Systems ◽  
Infinite Intervals ◽  
Defect Indices

This paper is concerned with formallyJ-self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All theJ-self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all theJ-self-adjoint subspace extensions are given in the limit point and limit circle cases.

Sign in / Sign up

Export Citation Format

Share Document