Order Non-autonomous Hamiltonian Systems Research Articles

Multiplicity of Periodic Solutions for Second Order Hamiltonian Systems with Asymptotically Quadratic Conditions

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper. Using Fountain Theorem, one multiplicity result of periodic solutions is obtained, which improves some previous results.

Subharmonic Pl-solutions of first order Hamiltonian systems

In this paper, for any symplectic matrix P, the existence of subharmonic Pl-solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct Pl-solutions is proved.

On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems

In this paper, we are concerned with the existence of periodic solutions for second order non-autonomous Hamiltonian systems under a new subquadratic growth condition. By using the minimax methods in critical point theory, an existence theorem is obtained, which extends and improves some known results in the literature.

Homoclinic Solutions for a Class of Second Order Nonautonomous 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.

Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales

In this paper, we are concerned with a second order non-autonomous Hamiltonian system on time scales TuΔΔ(ρ(t))+Vu(t,u(t))=f(t),t∈Tκ. Under certain conditions, the existence and multiplicity of periodic solutions are obtained for this Hamiltonian system on time scales by using the saddle point theory, the least action principle as well as the three-critical-point theorem. In addition, the existence of homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a given sequence of Hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument.

THE EXISTENCE OF HETEROCLINIC ORBITS FOR A SECOND ORDER HAMILTONIAN SYSTEM

In this paper, via variational methods and critical point theory, we study the existence of heteroclinic orbits for the following second order nonautonomous Hamiltonian system $$ \ddot{u}-\triangledown{F(t,u)} =0, $$ where $ u\in R^{n}$ and $ F\in C^{1}(R\times R^{n}, R), F\geq 0.$ $ \mathcal{M} \subset R^{n} $ be set of isolated points and $\sharp{\mathcal{M}}\geq 2.$ For each $\xi \in \mathcal{M},$ there exists a positive number $\rho_{0}$ such that if $y\in B_{\rho_{0}}(\xi),$ then $F(t, y)\geq F(t, \xi) $for all $t\in R$, where $B_{\rho_{0}}(\xi)=\{y\in R^{n}\mid \mid y-\xi \mid<\rho_{0}\}.$Under some more assumptions on $F(t, x)$ and $\mathcal{M}$, we prove that each point in $\mathcal{M}$ is joined to another point in $\mathcal{M}$ by a solution of our system.

A non-autonomous Hamiltonian system on time scales

In this paper we are concerned with a second order non-autonomous Hamiltonian system on time scales T of the form {uΔ2(t)=∇F(σ(t),uσ(t)),Δ-a.e.t∈[0,T]Tκ,u(0)−u(T)=0,uΔ(0)−uΔ(T)=0. Some results on the existence and multiplicity of periodic solutions are obtained for the Hamiltonian system by means of the saddle point theorem, the least action principle as well as the three-critical-point theorem.

Brake subharmonic solutions of first order Hamiltonian systems

In this paper, we mainly use the Galerkin approximation method and the iteration inequalities of the $L$-Maslov type index theory to study the properties of brake subharmonic solutions for the first order non-autonomous Hamiltonian systems. We prove that when the positive integers $j$ and $k$ satisfies the certain conditions, there exists a $jT$-periodic nonconstant brake solution $z_{j}$ such that $z_{j}$ and $z_{kj}$ are distinct.

Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems

In this paper we are devoted to considering the existence of homoclinic solutions for some second order non-autonomous Hamiltonian systems with the asymptotically quadratic potential at infinity. The proof is based on a variant version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.

New existence of periodic solutions for second order non-autonomous Hamiltonian systems

By using mountain pass theorem and local link theorem, some existence theorems are obtained for periodic solutions of second order non-autonomous Hamiltonian systems under local superquadratic condition and other suitable conditions.

Periodic solutions of a class of second order non-autonomous Hamiltonian systems

Some existence theorems are obtained for periodic solutions of non-autonomous second order systems by using the least action principle and minimax methods in critical point theory. Our results extend and improve many previously known results.

Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition

In this paper we prove the existence of homoclinic orbits for thefirst order non-autonomous Hamiltonian system$\dot{z}=\mathcal {J}H_{z}(t,z),$where $H(t,z)$ depends periodically on $t$. We establish someexistence results of the homoclinic orbits for weak superlinearcases. To this purpose, we apply a new linking theorem to providebounded Palais-Samle sequences.

Homoclinic solutions for some second order non-autonomous Hamiltonian systems without the globally superquadratic condition

This paper deals with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian system (HS) q ̈ + V q ( t , q ) = f ( t ) , where V ∈ C 1 ( R × R n , R ) , V ( t , q ) = − K ( t , q ) + W ( t , q ) is T -periodic in t , f is aperiodic and belongs to L 2 ( R , R n ) . Under the assumptions that K satisfies the “pinching” condition b 1 | q | 2 ≤ K ( t , q ) ≤ b 2 | q | 2 , W ( t , q ) is not globally superquadratic on q and some additionally reasonable assumptions, we give a new existence result to guarantee that (HS) has a homoclinic solution q ( t ) emanating from 0 . The homoclinic solution q ( t ) is obtained as a limit of 2 k T -periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the Mountain Pass Theorem. Recent results in the literature are generalized and significantly improved.

Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential

Some existence theorems are obtained for periodic solutions of a class of non-autonomous second order Hamiltonian systems with local superquadratic potential, by making use of the generalized mountain pass theorem.

Periodic Solutions for Second-Order Differential Equations involving Nonconvex Superpotentials

The paper establishes the existence of a nonconstant periodic solution of a general second order nonautonomous Hamiltonian system with discontinuous nonlinearities. The multiplicity of solutions is also studied.

Periodic and homoclinic solutions for a class of unilateral problems

This paper contains some existence and multiplicity results for periodic solutions of second order nonautonomous and nonsmooth Hamiltonian systems involving nonconvex superpotentials. This study is achieved by proving the existence of homoclinic solutions. These solutions are constructed as critical points of the corresponding nonconvex and nonsmooth energy functional.