Existence of infinitely many homoclinic orbits in Hamiltonian systems

2011 ◽  
Vol 141 (5) ◽  
pp. 1103-1119 ◽  
Author(s):  
X. H. Tang ◽  
Xiaoyan Lin
Keyword(s):  
Potential Function ◽  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Symmetric Mountain Pass Theorem ◽  
Second Order Hamiltonian Systems ◽  
Existence Criteria

By using the symmetric mountain pass theorem, we establish some new existence criteria to guarantee that the second-order Hamiltonian systems ü(t) − L(t)u(t) + ∇W(t,u(t)) = 0 have infinitely many homoclinic orbits, where t ∈ ℝ, u ∈ ℝN, L ∈ C(ℝ, ℝN × N) and W ∈ C1(ℝ × ℝN, ℝ) are not periodic in t. Our results generalize and improve some existing results in the literature by relaxing the conditions on the potential function W(t, x).

scholarly journals Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ying Lv ◽  
Chun-Lei Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Mountain Pass ◽  
Fountain Theorem ◽  
Existence And Multiplicity ◽  
Second Order Hamiltonian Systems ◽  
Superquadratic Potential

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

Existence of homoclinic solutions for a class of second-order non-autonomous Hamiltonian systems

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Qiongfen Zhang ◽  
X. Tang
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Second Order Hamiltonian Systems

AbstractBy using the variant version of Mountain Pass Theorem, the existence of homoclinic solutions for a class of second-order Hamiltonian systems is obtained. The result obtained generalizes and improves some known works.

Infinitely-many solutions for subquadratic fractional Hamiltonian systems with potential changing sign

2015 ◽  
Vol 4 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Ziheng Zhang ◽  
Rong Yuan
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Mountain Pass ◽  
Infinitely Many Solutions ◽  
Fractional Hamiltonian Systems ◽  
Symmetric Mountain Pass Theorem

AbstractIn this paper we are concerned with the existence of infinitely-many solutions for fractional Hamiltonian systems of the form ${\,}_tD^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}u(t))+L(t)u(t)=\nabla W(t,u(t))$, where ${\alpha \in (\frac{1}{2},1)}$, ${t\in \mathbb {R}}$, ${u\in \mathbb {R}^n}$, ${L\in C(\mathbb {R},\mathbb {R}^{n^2})}$ is a symmetric and positive definite matrix for all ${t\in \mathbb {R}}$, ${W\in C^1(\mathbb {R}\times \mathbb {R}^n,\mathbb {R})}$ and ${\nabla W(t,u)}$ is the gradient of ${W(t,u)}$ at u. The novelty of this paper is that, assuming L(t) is bounded in the sense that there are constants ${0<\tau _1<\tau _2< \infty }$ such that ${\tau _1 |u|^2\le (L(t)u,u)\le \tau _2 |u|^2}$ for all ${(t,u)\in \mathbb {R}\times \mathbb {R}^n}$ and ${W(t,u)}$ is of the form ${({a(t)}/({p+1}))|u|^{p+1}}$ such that ${a\in L^{\infty }(\mathbb {R},\mathbb {R})}$ can change its sign and ${0<p<1}$ is a constant, we show that the above fractional Hamiltonian systems possess infinitely-many solutions. The proof is based on the symmetric mountain pass theorem. Recent results in the literature are generalized and significantly improved.

scholarly journals Homoclinic Solutions for a Class of the Second-Order Impulsive Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jingli Xie ◽  
Zhiguo Luo ◽  
Guoping Chen
Keyword(s):  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Solution ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Approximation Solution ◽  
Periodic Approximation

This paper is concerned with the existence of homoclinic solutions for a class of the second order impulsive Hamiltonian systems. By employing the Mountain Pass Theorem, we demonstrate that the limit of a2kT-periodic approximation solution is a homoclinic solution of our problem.

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