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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.

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.

In this paper, we study the multiplicity of periodic solutions for the second order Hamiltonian systems u ¨ + ∇ F (t , u) = 0 with the boundary condition u (0) - u (T) = u ˙ (0) - u ˙ (T) = 0 , where the potential F is either subquadratic k(t)-concave or subquadratic μ (t) -convex. Based on the reduction method and a three-critical-point theorem due to Brezis and Nirenberg, we obtain the multiplicity results, which complement and sharply improve some related results in the literature. [ABSTRACT FROM AUTHOR]

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.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

This paper investigates solutions for subquadratic convex or $B$-concave operator equations. First, some existence results are obtained by the index theory and the critical point theory. Then, some applications to second order Hamiltonian systems satisfying generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions are pointed out. In particular, some well known theorems about periodic solutions for second order Hamiltonian systems are special cases of these results.

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.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

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. [ABSTRACT FROM AUTHOR]

We look for homoclinic solutions q : R RN to the class of second order Hamiltonian systems ... where L : R ! RN×N and a, b : R ! R are positive bounded functions, G1,G2 : RN ! R are positive homogeneous functions and f : R ! RN. Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if f 0 and the existence of at least three solutions if f is not trivial but small enough. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: \begin{equation} \begin{cases} \ddot u(t)+m^2\omega^2u(t)+\nabla F(t,u(t))=0\qquad \mbox{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0, \end{cases} \end{equation} where $m>0$, the potential $F(t,x)$ is convex in $x$ and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem [Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989].

In this paper we consider a class of super-linear second order Hamiltonian systems. We use Morse theory to obtain the existence and multiplicity of rotating periodic solutions, which might be periodic, subharmonic or quasi-periodic ones. [ABSTRACT FROM AUTHOR]

We consider a class of second order Hamiltonian systems with a $C^{2}$ potential function. The existence of new periodic solutions with a prescribed energy is established by the use of constrained variational methods. [ABSTRACT FROM AUTHOR]

In this paper we present some recent multiplicity results for a class of second order Hamiltonian systems. Exploiting the variational structure of the problem, it will be shown how the existence of multiple, even infinitely many, periodic solutions can be assured.

In this paper, we study the existence of periodic solutions with prescribed minimal period for even superquadratic autonomous second order Hamiltonian systems defined on R n with no convexity assumptions. We use a direct variational approach for this problem on a W 1, 2 space of functions invariant under the action of a transformation group isomorphic to the Klein Fourgroup V 4 = Z 2 ⊕ Z 2 to find symmetric periodic solutions, and prove a new iteration inequality on the Morse index by iterating such functions properly. Using these tools and the Mountain-pass theorem, we show that for every T > 0 the abobe mentioned system possesses a T-periodic solution x ( t ) with minimal period T or T/3, and this solution is even about t = 0, T/2 and odd about t = T/4, 3 T/4. [ABSTRACT FROM AUTHOR]

Using the least action principle, the existence of periodic solutions for some nonautonomous second order Hamiltonian systems is obtained. Using minimax methods, the multiplicity of periodic solutions is obtained. Our results extend some previous results. [ABSTRACT FROM AUTHOR]

Multiplicity results for an eigenvalue second-order Hamiltonian system are investigated. Using suitable critical points arguments, the existence of an exactly determined open interval of positive eigenvalues for which the system admits at least three distinct periodic solutions is established. Moreover, when the energy functional related to the Hamiltonian system is not coercive, an existence result of two distinct periodic solutions is given.

Under some local superquadratic conditions on $W(t,u)$ with respect to u, the existence of infinitely many homoclinic solutions is obtained for the nonperiodic second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0$, $\forall t\in\mathbb{R}$, where $L(t)$ is unnecessarily coercive. [ABSTRACT FROM AUTHOR]

Motivated by the instability of suspension bridges, we consider a class of second order Hamiltonian systems where one component initially holds almost all the energy of the system. We show that if the total energy is sufficiently small then it remains on this component, whereas if the total energy is larger it may transfer to the other components. Through Mathieu equations we explain the precise mechanism which governs the energy transfer. [ABSTRACT FROM AUTHOR]

By using the least action principle and the minimax methods, the existence of periodic solutions for a class of second order Hamiltonian systems is considered. The results obtained in this paper extend some previous results. [ABSTRACT FROM AUTHOR]

Abstract: A new superquadratic growth condition is introduced, which is an extension of the well-known superquadratic growth condition due to P.H. Rabinowitz and the nonquadratic growth condition due to Gui-Hua Fei. An existence theorem is obtained for periodic solutions of a class of new superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, specially, a local linking theorem. [Copyright &y& Elsevier]

Some existence theorems are obtained for periodic solutions of second order Hamiltonian system by using the minimax principle. Our results improve those in some known literatures. [ABSTRACT FROM AUTHOR]

By using the least action principle and minimax methods in critical point theory, some existence theorems for periodic solutions of second order Hamiltonian systems are obtained. [ABSTRACT FROM AUTHOR]

Abstract: The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systems , by the local linking lemma, where is superquadratic and need not be nonnegative globally. [Copyright &y& Elsevier]

Abstract: Some critical point theorems without the compactness assumptions are obtained by the reduction method, the perturbation argument and the least action principle. As applications, some existence results of periodic solutions are obtained for nonautonomous second order Hamiltonian systems, which unify and generalize some recent corresponding results. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence of three distinct solutions for the following problem where , T is a real positive number, is a continuous map from the interval to the set of N-order symmetric matrices. We propose sufficient conditions only on the potential F. More precisely, we assume that G satisfies only a usual growth condition which allows us to use a variational approach. [Copyright &y& Elsevier]

Abstract: In this paper, using well-known variational methods, we prove a general abstract theorem on the existence of a critical point for resonant problems with unbounded nonlinearities and then apply it to ordinary and partial differential equations with generalized Ahmad–Lazer–Paul conditions. The abstract result contains several concrete results in the literature and can also be used to deal with some new cases for resonant differential equations. [Copyright &y& Elsevier]

Multiplicity results for an eigenvalue second-order Hamiltonian system are investigated. Using suitable critical points arguments, the existence of an exactly determined open interval of positive eigenvalues for which the system admits at least three distinct periodic solutions is established. Moreover, when the energy functional related to the Hamiltonian system is not coercive, an existence result of two distinct periodic solutions is given. [ABSTRACT FROM AUTHOR]

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems u¨(t)−L(t)u(t)+∇W(t,u(t))=0, ∀t∈R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t∈R, and W(t,x) satisfies the superquadratic condition W(t,x)/&z.sfnc;x&z.sfnc;2→+∞ as &z.sfnc;x&z.sfnc;→∞ uniformly in t, and need not satisfy the global Ambrosetti–Rabinowitz condition. [Copyright &y& Elsevier]