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.

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.

scholarly journals Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ziheng Zhang ◽  
Fang-Fang Liao ◽  
Patricia J. Y. Wong
Keyword(s):  
Hamiltonian Systems ◽  
Bounded Function ◽  
Homoclinic Solution ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Global Maximum ◽  
Continuous Bounded Function ◽  
Almost Periodic ◽  
Strong Force ◽  
Singular Hamiltonian Systems

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systemsu¨+atWuu=0, (HS) where-∞<t<+∞,u=u1,u2, …,uN∈ℝNN≥3,a:ℝ→ℝis a continuous bounded function, and the potentialW:ℝN∖{ξ}→ℝhas a singularity at0≠ξ∈ℝN, andWuuis the gradient ofWatu. The novelty of this paper is that, for the case thatN≥3and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum ofW. Different from the cases that (HS) is autonomousat≡1or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous andN≥3. Besides the usual conditions onW, we need the assumption thata′t<0for allt∈ℝto guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

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.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Qiongfen Zhang ◽  
Yuan Li
Keyword(s):  
Continuous Function ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Space ◽  
Point Theory ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonautonomous Systems ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity

This paper is concerned with the following nonlinear second-order nonautonomous problem:ü(t)+q(t)u̇(t)-∇K(t,u(t))+∇W(t,u(t))=0, wheret∈R,u∈RN, andK,W∈C1(R×RN,R)are not periodic intandq:R→Ris a continuous function andQ(t)=∫0t‍q(s)dswithlim|t|→+∞⁡Q(t)=+∞. The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.

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 Homoclinic Solutions for a Second-Order Nonperiodic Asymptotically Linear Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Qiang Zheng
Keyword(s):  
Boundary Value Problems ◽  
Hamiltonian Systems ◽  
Existence Result ◽  
Boundary Value ◽  
Homoclinic Solution ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Asymptotically Linear ◽  
Minimax Methods ◽  
Linear Hamiltonian Systems

We establish a new existence result on homoclinic solutions for a second-order nonperiodic Hamiltonian systems. This homoclinic solution is obtained as a limit of solutions of a certain sequence of nil-boundary value problems which are obtained by the minimax methods. Some recent results in the literature are generalized and extended.

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