Homoclinic Solutions for a Class of Second Order Hamiltonian Systems

In this paper we consider the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system $${\ddot q}-L(t)q+\nabla W(t,q)=0, \quad\quad\quad\quad\quad\quad\quad (\rm HS)$$ where ${L\in C({\mathbb R},{\mathbb R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in {\mathbb R}}$ , W(t, q)?=?a(t)U(q) with ${a\in C({\mathbb R},{\mathbb R}^+)}$ and ${U\in C^1({\mathbb R}^n,{\mathbb R})}$ . The novelty of this paper is that, assuming L is bounded from below in the sense that there is a constant M?>?0 such that (L(t)q, q)?≥ M |q|² for all ${(t,q)\in {\mathbb R}\times {\mathbb R}^n}$ , we establish one new compact embedding theorem. Subsequently, supposing that U satisfies the global Ambrosetti–Rabinowitz condition, we obtain a new criterion to guarantee that (HS) has one nontrivial homoclinic solution using the Mountain Pass Theorem, moreover, if U is even, then (HS) has infinitely many distinct homoclinic solutions. Recent results from the literature are generalized and significantly improved.     

Multiple of Homoclinic Solutions for a Perturbed Dynamical Systems with Combined Nonlinearities

In this paper, we study the existence of multiple and infinite homoclinic solutions for the following perturbed dynamical systems

$$\begin{aligned} \ddot{x}+A\cdot \dot{x}-L(t)\cdot x+\nabla W(t,x)=f(t), \end{aligned}$$

where \(t\in {\mathbb R}, x\in {\mathbb R}^N,\) A is an antisymmetric constant matrix, the matrix L(t) is not necessary positive definite for all \(t\in {\mathbb R}\) nor coercive, the nonlinearity \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) involves a combination of superquadratic and subquadratic terms and is allowed to be sign-changing and \(f\in C({\mathbb R},{\mathbb R}^{N})\cap L^{2}({\mathbb R},{\mathbb R}^{N}).\) Recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.

     

二阶离散Hamiltonian系统的多重周期解

利用变分原理和Clark定理,研究了带参数的二阶离散Hamiltonian系统的多重周期解,得到了此类方程周期解个数的下界估计.     

Variational Approach to Homoclinic Solutions for Fractional Hamiltonian Systems

In this paper, we discuss the existence and multiplicity of homoclinic solutions for fractional Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Sufficient conditions ensuring the existence of an unbounded sequence of homoclinic solutions for the given problem are obtained via variational approach.     

一类有小扰动项的二阶哈密尔顿系统的多重解的存在性

基于变分法,运用三临界点定理,得到了一类有小扰动项的二阶哈密尔顿系统的多重解的存在性,推广了已有文献的相关结果.     

Multiplicity of Homoclinic Solutions for Second Order Hamiltonian Systems with Local Conditions at the Origin

In this paper,the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systems ü(t)-L(t)u(t)+▽_uW(t,u(t))=0 is obtained via a new Symmetric Mountain Pass Lemma established by Kajikiya,where L ∈C(R,R^N×N) is symmetric but non-periodic,W ∈C¹(R×R^N,R)is locally even in u and only satisfies some growth conditions near u=0,which improves some previous results.     

一类二阶渐近周期Hamilton系统同宿轨道

运用变分方法讨论二阶渐近周期Hamilton系统 -u+L(t)u=(1+g(t))V′(t,u)的Lagrange泛函在流形上的极小问题,进而证明该系统存在非平凡同宿轨道,其中L,V关于t是周期的,g(t)→0(|t|→∞).     

一类弱二次Hamilton系统的同宿解的多重性结果

考虑二阶非自治弱二次Hamilton系统同宿解的多重性.一般考虑的势函数关于u在无穷远点处的下界函数是一个正常数.而当该系数换为一个正函数而非常数时,情况就会相当不同.该文中讨论了这一问题,将此系数推广到下确界可以是0的一个有关t的正函数.因此该文的结果是对近期一些结果的有意义的改进.     

对称超二次二阶哈密尔顿系统的周期解

我们利用Ambrosetti-Rabinowitz对称形式的山路引理证明了给定周期T的对称超二次二阶哈密尔顿系统具有无穷多个反T/2-周期且奇的周期解.     

Homoclinic Solutions for A Class of Second Order Neutral Functional Differential Systems

By means of an extension of Mawhin's continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.     

Periodic Solutions of Nonautonomous Second Order Hamiltonian Systems

In this paper, we develop the local linking theorem given by Li and Willein by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems （H） ü＋A（t）u＋∨V（t, u）=0, u∈R^N, t∈R. We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem.     

Homoclinic Orbits for Second Order Discrete Hamiltonian Systems with Potential Changing Sign

In this paper, a class of second order discrete Hamiltonian systems without any periodicity assumptions are considered. Base on the critical point theory, some sufficient conditions for the existence of homoclinic orbits are obtained. The results obtained extend the results in [2006] by relaxing the assumptions on the sign of the potential.      

一类二阶哈密尔顿系统的周期解

在(CPS)_C及(PS)_C条件下,利用Ambrosetti-Rabinowitz对称形式的山路引理,研究了一类二阶哈密尔顿保守系统在给定能量面上的无穷多个周期解的存在性问题.     

Lyapunov–Schmidt Approach to Studying Homoclinic Splitting in Weakly Perturbed Lagrangian and Hamiltonian Systems

We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems.     

具有小受迫项的哈密顿系统的同宿轨道

By Brezis-Nirenberg type Mountain Pass Theorem, the research has focused on the existence of nontrivial homoclinic orbits for a class of second order Hamiltonian systems with non-Ambrosetti-Rabinowitz type superquadratic potentials and small forced terms.     

Slow Evolution in Perturbed Hamiltonian Systems

For parametrized Hamiltonian systems with an arbitrary, finite number of degrees of freedom, it is shown that secularly stable families of equilibrium solutions represent approximate trajectories for small (not necessarily Hamiltonian) perturbations of the original system. This basic result is further generalized to certain conservative, but not necessarily Hamiltonian, systems of differential equations. It generalizes to the conservative case a theorem due, in the dissipative case, to Tikhonov, to Gradshtein, and to Levin and Levinson. It justifies the use of physically motivated approximation procedures without invoking the method of averaging and without requiring nonresonance conditions or the integrability of the unperturbed Hamiltonian.     

离散哈密顿系统多个周期解的存在性

本文利用临界点理论,建立了一类离散哈密顿系统存在多个周期解的一些充分条件.     

一类二阶离散哈密尔顿系统的周期解的多重性

基于变分法,运用三临界点定理,得到了一类有小强迫项的二阶离散哈密尔顿系统的周期解的多重性,推广了现有文献的相关结果,并给出例子加以说明.     

一类Hamilton系统的同宿解存在性

We study the existence of homoclinic orbits for some Hamiltonian system.A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a sequence of systems of differential equations.     

Multiplicity of Periodic Solutions for Second Order Hamiltonian Systems with Asymptotically Quadratic Conditions

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper.Using Fountain Theorem,one multiplicity result of periodic solutions is obtained,which improves some previous results.