Chapter 2 Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods

2006 ◽  
pp. 77-146 ◽  
Author(s):  
Thomas Bartsch ◽  
Andrzej Szulkin
Keyword(s):  
Variational Methods ◽  
Hamiltonian Systems ◽  
Homoclinic Solutions

scholarly journals Multiple homoclinic solutions for p-Laplacian Hamiltonian systems with concave–convex nonlinearities

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lili Wan
Keyword(s):  
Variational Methods ◽  
Hamiltonian Systems ◽  
Homoclinic Solutions

AbstractThe multiplicity of homoclinic solutions is obtained for a class of the p-Laplacian Hamiltonian systems $\frac{d}{dt}(|\dot{u}(t)|^{p-2}\dot{u}(t))-a(t)|u(t)|^{p-2}u(t)+ \nabla W(t,u(t))=0$ddt(|u˙(t)|p−2u˙(t))−a(t)|u(t)|p−2u(t)+∇W(t,u(t))=0 via variational methods, where $a(t)$a(t) is neither coercive nor bounded necessarily and $W(t,u)$W(t,u) is under new concave–convex conditions. Recent results in the literature are generalized even for $p=2$p=2.

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 Periodic Solutions for a Class of Singular Hamiltonian Systems on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiaofang Meng ◽  
Yongkun Li
Keyword(s):  
Variational Methods ◽  
Critical Point ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Time Scales ◽  
Critical Point Theory ◽  
Point Theory ◽  
Singular Hamiltonian Systems ◽  
Existence Of Periodic Solutions

We are concerned with a class of singular Hamiltonian systems on time scales. Some results on the existence of periodic solutions are obtained for the system under consideration by means of the variational methods and the critical point theory.

scholarly journals Variational Approach to Quasi-Periodic Solution of Nonautonomous Second-Order Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Juhong Kuang
Keyword(s):  
Periodic Solution ◽  
Variational Methods ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Variational Approach ◽  
Second Order ◽  
New Approach ◽  
Second Order Hamiltonian Systems

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systemsx¨(t)=∇F(t,x(t)), wherex(t)=(x1(t),…,xN(t)), and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

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