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In this paper, we prove the existence of infinitely many solutions for the following nonperiodic fractional Hamiltonian system $$\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in {\mathbb {R}}\\ u\in H^{\alpha }({\mathbb {R}}), \end{array}\right. \end{aligned}$$ where $$_{-\infty }D_{t}^{\alpha }$$ and $$_{t}D^{\alpha }_{\infty }$$ are left and right Liouville–Weyl fractional derivatives of order $$\frac{1}{2}

Consider the following fractional Hamiltonian system $$\begin{aligned} \left\{ \begin{array}{l} _{t}D_{\infty }^{\alpha }(_{-\infty }D_{t}^{\alpha }u)(t)+L(t)u(t)=\nabla W(t,u(t)),\ t\in {\mathbb {R}}\\ u\in H^{\alpha }({\mathbb {R}}), \end{array}\right. \qquad \qquad \qquad (1) \end{aligned}$$ where $$_{t}D_{\infty }^{\alpha }$$ and $$_{-\infty }D^{\alpha }_{t}$$ are the Liouville–Weyl fractional derivatives of order $$\frac{1}{2}

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in\mathbb{R} $$ where $q\in C(\mathbb{R},\mathbb{R})$, $L\in C(\mathbb{R},\mathbb{R}^{N^{2}})$ is a symmatric and positive definite matix-valued function and $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{N},\mathbb{R})$. The novelty of this paper is that, assuming that $L$ is bounded from below unnecessarily coercive at infinity, and $W$ is only locally defined near the origin with respect to the second variable, we show that $(1)$ possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

We study the existence of fast homoclinic orbits for the following damped vibration system $$\ddot{u}(t)+q(t)\dot{u}(t)+\nabla V(t,u(t))=0$$ ; where $$q\in C(\mathbb {R},\mathbb {R})$$ and $$V\in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})$$ is of the type V(t,x)=-K(t,x)+W(t,x). A map K is not a quadratic form in x and W(t, x) is superquadratic in x.

Consider the following damped vibration system $$\begin{aligned} \ddot{u}(t)+q(t)\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0,\ \forall t\in \mathbb {R} \qquad \qquad (1) \end{aligned}$$ where $$q\in C(\mathbb {R},\mathbb {R})$$ , $$L\in C(\mathbb {R},\mathbb {R}^{N^{2}})$$ and $$W\in C(\mathbb {R}\times \mathbb {R}^{N},\ \mathbb {R})$$ . Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many fast homoclinic solutions for (1) when L is not required to be either uniformly positive definite or coercive and W satisfies some general super-quadratic conditions at infinity in the second variable but does not satisfy the classical superquadratic growth conditions at infinity.

In the present paper, we consider the fourth-order differential equation $$\begin{aligned} u^{(4)}(x)+\omega u''(x)+a(x)u(x)=f(x,u(x)),\ \forall x\in {\mathbb {R}}\quad \quad \quad \quad (1) \end{aligned}$$ in which $$\omega $$ represents a constant, $$a\in C({\mathbb {R}},{\mathbb {R}})$$ and $$f\in C({\mathbb {R}}^{2},{\mathbb {R}})$$ . We are concerned with the existence of ground state homoclinic solution for (1) when a is unnecessary positive and $$F(x,u)=\int ^{u}_{0}f(x,t)dt$$ satisfies a kind of superquadratic conditions due to Ding and Luan. For the proof, we apply a variant generalized weak linking theorem developed by Schechter and Zou. Some results in the literature are generalized and improved.

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.

Applying a Symmetric Mountain Pass Theorem, we prove the existence of infinitely many homoclinic solutions for a class of fourth-order differential equations u(4)(x) + ωu00(x) + a(x)u(x) = f(x,u(x)), ∀x ∈R where a ∈ C(R,R) may be negative on a bounded interval and F(x,u) =Ru 0 f(x,t)dt is superquadratic at infinity in the second variable but does not need to satisfy the well-known Ambrosetti-Rabinowitz superquadratic growth condition

In this paper, we study the existence of even homoclinic solutions for a damped vibration system x ( t ) + q ( t ) x ( t ) + V ′ ( t , x ( t ) ) = 0 , where q is a continuously differentiable function and V ∈ C 1 ( R × R N , R ) , V ( t , x ) = − K ( t , x ) + W ( t , x ) . We use a new kind of superquadratic condition instead of the global Ambrosetti–Rabinowitz condition. For the proof we use the Mountain Pass Theorem.

In this paper, we are interested in the following damped vibration system: where B is an antisymmetric $$N\times N$$ constant matrix, $$q:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ is a continuous function, $$L(t)\in C({\mathbb {R}},{\mathbb {R}}^{N^{2}})$$ is a symmetric matrix, and $$W(t,x)\in C^{1}({\mathbb {R}}\times {\mathbb {R}}^{N},{\mathbb {R}})$$ are neither autonomous nor periodic in t. The novelty of this paper is that, supposing that $$Q(t)=\int ^{t}_{0}q(s)\mathrm{d}s$$ is bounded from below and L(t) is coercive unnecessarily uniformly positively definite for all $$t\in {\mathbb {R}}$$ , we establish the existence of ground-state homoclinic solutions for (1) when the potential W(t, x) satisfies a kind of superquadratic conditions due to Ding and Luan for Schr $${\ddot{o}}$$ dinger equation. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou. Some recent results in the literature are generalized and significantly improved.

In this paper, we prove the existence of homoclinic solutions for a class of noncoercive first order Hamiltonian systems J x − M ( t ) x + u ∗ G ′ ( t , u ( x ) ) = 0 , by the minimax methods in critical point theory, specially, a Generalized Mountain Pass Theorem, when u is a linear operator with adjoint u ∗ and G ( t , y ) satisfies the superquadratic condition G ( t , x ) | y | 2 ⟶ − + ∞ as | y | ⟶ ∞ , uniformly in t , and need not satisfy the global Ambrosetti–Rabinowitz condition.

Considering a decomposition R2N=A♁B of R2N , we prove in this work, the existence of at least (1+dimA) geometrically distinct periodic solutions for the first-order Hamiltonian system Jx'(t)+H'(t,x(t))+e(t)=0 when the Hamiltonian H(t,u+v) is periodic in (t,u) and its growth at infinity in v is at most like or faster than |v|a, 0≤a

In this work, we prove the existence of subharmonics of the Hamiltonian system x = J H ′ ( t , x ) when the Hamiltonian H is coercive or resonant in a part of the variables uniformly in the other part. We derive the existence of such solutions by using minimax theory.

In this paper, we prove the existence of subharmonic solutions with prescribed minimal period for the Hamiltonian system u ( t ) = J ∇ H ( t , u ( t ) ) when the Hamiltonian H is subquadratic with unbounded gradient and not uniformly coercive.

In this paper we study in some particular cases, the existence and multiplicity of closed trajectories of a relativistic particle moving in some electromagnetic fields. To solve this problem, we use Hamiltonian systems and variational methods.

In this paper we study the existence of subharmonic solu- tions of the Hamiltonian system J ˙ x + u ∗ ∇G(t,u(x)) = e(t) where u is a linear map, G is a C 1 -function and e is a continuous function.

— The existence of subharmonic and multiple periodic solutions as well as the minimality of periods are obtained for the nonautonomous Hamiltonian systems ẋ = JH ′(t, x) with locally and partially sublinear Hamiltonian H; that is, there exist a decomposition IR = A ⊕ B of IR , an α ∈]0, 1[ and two periodic functions a ∈ L 2 1−α ([0, T ], IR) and b ∈ L([0, T ], IR) such that |H ′(t, u+ v)| ≤ a(t) |v| + b(t) for all (t, u, v) ∈ [0, T ] × A × B and H(t,u+v) |v|2α −→ +∞ or − ∞ as |v| −→ ∞ in B, uniformly in u ∈ A for a.e. t in some non empty open subset C of [0, T ]. For the resolution we use an analogy of Egorov’s theorem and a generalized saddle point theorem.

In this work, we prove the existence of multiple periodic and subharmonic solutions of the Hamiltonian system

Nous étudions les solutions du problème différentiel, Timoumi Mohsen. Solutions périodiques de systèmes hamiltoniens convexes non coercitifs. In: Bulletin de la Classe des sciences, tome 75, 1989. pp. 463-481.

In this paper, the existence of subharmonic solutions for a class of non-autonomous first-order Hamiltonian systems is investigated. We also study the minimality of periods for such solutions. Our results which extend and improve many previous results will be illustrated by specific examples. Our main tools are the minimax methods in critical point theory and the least action principle. {\bf Key words.} Hamiltonian systems. Critical point theory. Least action principle. Subharmonic solutions., Comment: 17 pages