scholarly journals HOMOCLINIC ORBITS FOR THE FIRST-ORDER HAMILTONIAN SYSTEM WITH SUPERQUADRATIC NONLINEARITY

2015 ◽  
Vol 19 (3) ◽  
pp. 673-690 ◽  
Author(s):  
Wen Zhang ◽  
Xianhua Tang ◽  
Jian Zhang
Keyword(s):  
Hamiltonian System ◽  
Homoclinic Orbits ◽  
First Order ◽  
Order Hamiltonian System ◽  
Superquadratic Nonlinearity

scholarly journals Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Wenping Qin ◽  
Jian Zhang ◽  
Fukun Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Point Spectrum ◽  
Homoclinic Orbits ◽  
Hamiltonian Operator ◽  
Quadratic Nonlinearity ◽  
Linking Theorem ◽  
Strongly Indefinite Functionals ◽  
Superquadratic Nonlinearity

We study the following nonperiodic Hamiltonian systemż=JHz(t,z), whereH∈C1(R×R2N,R)is the formH(t,z)=(1/2)B(t)z⋅z+R(t,z). We introduce a new assumption onB(t)and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

scholarly journals Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Xiaoyan Lin ◽  
Qi-Ming Zhang ◽  
X. H. Tang
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Satisfying Condition ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
First Order ◽  
Nonlinear Hamiltonian System

We give several sufficient conditions under which the first-order nonlinear Hamiltonian systemx'(t)=α(t)x(t)+f(t,y(t)),  y'(t)=-g(t,x(t))-α(t)y(t)has no solution(x(t),y(t))satisfying condition0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍,whereμ,ν>1and(1/μ)+(1/ν)=1,0≤xf(t,x)≤β(t)|x|μ,xg(t,x)≤γ0(t)|x|ν,β(t),γ0(t)≥0, andα(t)are locally Lebesgue integrable real-valued functions defined onℝ.

Homoclinic orbits for a class of Hamiltonian systems

1990 ◽  
Vol 114 (1-2) ◽  
pp. 33-38 ◽  
Author(s):  
Paul H. Rabinowitz
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Mountain Pass Theorem ◽  
Local Maximum ◽  
Homoclinic Orbits ◽  
Order Hamiltonian System ◽  
Image Position ◽  
Second Order Hamiltonian System

SynopsisConsider the second order Hamiltonian system:where q ∊ ℝn and V ∊ C1 (ℝ ×ℝn ℝ) is T periodic in t. Suppose Vq (t, 0) = 0, 0 is a local maximum for V(t,.) and V(t, x) | x| → ∞ Under these and some additional technical assumptions we prove that (HS) has a homoclinic orbit q emanating from 0. The orbit q is obtained as the limit as k → ∞ of 2kT periodic solutions (i.e. subharmonics) qk of (HS). The subharmonics qk are obtained in turn via the Mountain Pass Theorem.

Odd Homoclinic Orbits for a Second Order Hamiltonian System

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Liliane A. Maia ◽  
Olimpio H. Miyagaki ◽  
Sergio H. M. Soares
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Second Order ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Order Hamiltonian System ◽  
Compactness Principle ◽  
Second Order Hamiltonian System

AbstractThe aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

scholarly journals Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Long Yuhua
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Existence Results ◽  
Mountain Pass ◽  
First Order ◽  
Discrete Hamiltonian System ◽  
Generalized Mountain Pass Theorem

A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti-Rabinowitz condition.

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