Order Hamiltonian Systems Research Articles

Multiple Normalized Solutions for First Order Hamiltonian Systems

Multiple Normalized Solutions for First Order Hamiltonian Systems

Existence and concentration of homoclinic orbits for first order Hamiltonian systems

<abstract><p>This paper is concerned with the following first-order Hamiltonian system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \nonumber \dot{z} = \mathscr{J}H_{z}(t, z), \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where the Hamiltonian function $ H(t, z) = \frac{1}{2}Lz\cdot z+A(\epsilon t)G(|z|) $ and $ \epsilon > 0 $ is a small parameter. Under some natural conditions, we obtain a new existence result for ground state homoclinic orbits by applying variational methods. Moreover, the concentration behavior and exponential decay of these ground state homoclinic orbits are also investigated.</p></abstract>

Note on the Indices of the Discrete and Continuous Second Order Hamiltonian System

Note on the Indices of the Discrete and Continuous Second Order Hamiltonian System

Homoclinic Solutions Near the Origin for a Class of First Order Hamiltonian Systems

Homoclinic Solutions Near the Origin for a Class of First Order Hamiltonian Systems

Periodic and subharmonic solutions for a class of superquadratic first order Hamiltonian systems

Periodic and subharmonic solutions for a class of superquadratic first order Hamiltonian systems

Notes on Multiple Periodic Solutions for Second Order Hamiltonian Systems

Notes on Multiple Periodic Solutions for Second Order Hamiltonian Systems

Multiplicity of Homoclinic Solutions for Second Order Hamiltonian Systems with Local Conditions at the Origin

Multiplicity of Homoclinic Solutions for Second Order Hamiltonian Systems with Local Conditions at the Origin

On the existence of periodic solutions to second order Hamiltonian systems

In this paper, the existence of periodic solutions to the second order Hamiltonian systems is investigated. By introducing a new growth condition which generalizes the Ambrosetti–Rabinowitz condition, we prove a existence result of nontrivial $T$-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the potential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions.

Homoclinic orbits for periodic second order Hamiltonian systems with superlinear terms

We obtain the existence of nontrivial homoclinic orbits for nonautonomous second order Hamiltonian systems by using critical point theory under some different superlinear conditions from those previously used in Hamiltonian systems. In particular, an example is given to illustrate our result.

Multiplicity Of Periodic Solutions For Second Order Hamiltonian Systems With Mixed Nonlinearities

The multiplicity of periodic solutions for a class of second order Hamiltonian system with superquadratic plus subquadratic nonlinearity is studied in this paper. Obtained via the Symmetric Mountain Pass Lemma, two results about infinitely many periodic solutions of the systems extend some previously known results.

Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth

In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems.

A unified approach to periodic solutions for a class of non-autonomous second order Hamiltonian systems

This paper is dedicated to studying the periodic solutions of non-autonomous second order Hamiltonian systems. By virtue of control functions and the minimax methods in critical point theory, we propose a unified approach when the nonlinearity is unbounded while grows no more than |x| at infinity. We establish some new existence theorems on periodic solutions, which unify and generalize many previously known results in the literature.

Multiple solutions for super-linear symmetric operator equations

ABSTRACT This paper investigates multiple solutions for super-linear symmetric operator equations by optimization methods like index theory, Mountain pass theorem and mini-max methods. Applying the results to second order Hamiltonian systems with symmetry yields finite or infinitely many solutions.

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough.

Existence of solutions for subquadratic convex or $B$-concave operator equations and applications to second order Hamiltonian systems

Existence of solutions for subquadratic convex or $B$-concave operator equations and applications to second order Hamiltonian systems

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.

Periodic and Subharmonic Solutions for a Class of Sublinear First-Order Hamiltonian Systems

By the aids of variational methods, we study the existence of periodic and subharmonic solutions, and the minimality of their periods, for the nonautonomous first order Hamiltonian system $$J{\dot{u}}(t)+\nabla H(t,u(t))+h(t)=0$$ under some kind of sublinear nonlinearity conditions. Our results are new and generalize and extend some results in the literature.

Two homoclinic orbits for some second-order Hamiltonian systems

This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in \cite{IJ}.

On the Existence of Homoclinic Type Solutions of a Class of Inhomogenous Second Order Hamiltonian Systems

We show the existence of homoclinic type solutions of second order Hamiltonian systems of the type ddot{q}(t)+nabla _{q}V(t,q(t))=f(t), where tin mathbb {R}, the C^1-smooth potential V:mathbb {R}times mathbb {R}^nrightarrow mathbb {R} satisfies a relaxed superquadratic growth condition, its gradient is bounded in the time variable, and the forcing term f:mathbb {R}rightarrow mathbb {R}^n is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger time-periods. We prove that the latter systems admit periodic solutions of mountain-pass type, and obtain homoclinic type solutions of the original system from them by passing to the limit (in the topology of almost uniform convergence) when the periods go to infinity.

Subharmonic solutions for a class of Lagrangian systems

We prove that second order Hamiltonian systems \begin{document}$ -\ddot{u} = V_{u}(t,u) $\end{document} with a potential \begin{document}$ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $\end{document} of class \begin{document}$ C^1 $\end{document} , periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [ 14 ]. Indeed, we weaken the latter condition in a neighbourhood of \begin{document}$ 0\in \mathbb{R} ^N $\end{document} . We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.