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.

Linear second order elliptic equations with Venttsel boundary conditions

1991 ◽  
Vol 118 (3-4) ◽  
pp. 193-207 ◽  
Author(s):  
Yousong Luo ◽  
Neil S. Trudinger
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Existence Result ◽  
Boundary Value ◽  
Second Order ◽  
Classical Solutions ◽  
Schauder Estimate ◽  
Second Order Elliptic Equations

SynopsisWe prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.

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 Weyl-Titchmarsh Theory for Hamiltonian Dynamic Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-18 ◽  
Author(s):  
Shurong Sun ◽  
Martin Bohner ◽  
Shaozhu Chen
Keyword(s):  
Boundary Value Problems ◽  
Spectral Theory ◽  
Hamiltonian Systems ◽  
Dynamic Systems ◽  
Time Scale ◽  
Boundary Value ◽  
Hamiltonian Dynamic ◽  
Special Cases ◽  
Linear Hamiltonian Systems ◽  
Singular Hamiltonian Systems

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale&#x1D54B;, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for&#x1D54B;=ℝand&#x1D54B;=ℤwithin one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i)M(λ)theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

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.

scholarly journals An Existence Result of Positive Solutions for Fully Second-Order Boundary Value Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yongxiang Li ◽  
Yaya Shang
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Existence Result ◽  
Boundary Value ◽  
Second Order ◽  
Upper Solution ◽  
Superlinear Growth ◽  
Lower And Upper Solution

An existence result of positive solutions is obtained for the fully second-order boundary value problem  -u′′(t)=f(t,u(t),u′(t)),  t∈[0,1],  u(0)=u(1)=0,wheref:[0,1]×R+×R→Ris continuous. The nonlinearityf(t,x,y)may be sign-changing and superlinear growth onxandy. Our discussion is based on the method of lower and upper solution.

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