Fast homoclinic solutions for a class of damped vibration problems with subquadratic potentials

In this paper, we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problems where L(t) and W(t, x) are neither autonomous nor periodic in t. Our approach is variational and it is based on the critical point theory. We prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function. Our theorems appear to be the first such result and our results extend some recent works.     

Multiple homoclinic orbits for a class of Hamiltonian systems

In this paper, we obtain the existence of at least two nontrivial homoclinic orbits for a class of second order autonomous Hamiltonian systems. This multiplicity result is obtained by a new variational method based on the relative category: to overcome the lack of compactness of the problem, we first solve perturbed nonautonomous problems and study the limit of the solutions as the nonautonomous perturbation goes to 0. This method allows to get rid of some assumptions on the potential used in the work of Ambrosetti and Coti-Zelati. Received August 9, 1999 / Accepted September 7, 1999 / Published online September 14, 2000     

Infinitely many homoclinic solutions for a class of damped vibration problems

In this paper, we deal with the existence of infinitely many homoclinic solutions for the damped vibration problems where A is an antisymmetry N × N constant matrix, we establish some new existence results to guarantee that the above system has infinitely many homoclinic solutions under more relaxed assumptions on W(t,x), which satisfies a kind of new subquadratic condition by using fountain theorem. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity of homoclinic solutions for a class of damped vibration problems

The main purpose of this paper is to study the following damped vibration problems

(1.1)     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

On the non-existence of homoclinic orbits for a class of infinite dimensional Hamiltonian systems

We prove that for a class of infinite dimensional Hamiltonian systems certain homoclinic connections to the origin cease to exist when the non-linearities have `super-critical' growth. The proof is based on a variational principle and a Poho\v{z}aev type identity.

     

Existence of infinitely many homoclinic orbits for a class of aperiodic systems via variational methods

By applying Symmetric Mountain Pass Theorem in critical point theory, the existence of infinitely many homoclinic solutions is obtained for the following aperiodic system $$\frac{d}{dt}(A(t)\dot{u}(t))-B(t)u(t)+\nabla W(t,u(t))=0,$$ where t???, u??? ^N , A,B:???? ^N×N and W:?×? ^N ??? are not periodic in t.     

Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.     

Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems

We consider the Hamiltonian system q=L(t)q–V(t, q) in R ^m ,L and V being asymptotic, as t–, to certain periodic functions L_, V_. Under suitable assumptions on the functions L, L_, V, V_, we prove for any kN, the existence of infinitely many k- bump homoclinic solutions of the Hamiltonian system.     

On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems

We prove the existence of infinitely many homoclinic orbits on a Riemannian manifold (possibly non-compact), for a class of second order Hamiltonian systems of the form: $$D_t \dot x(t) + grad_x V(t,x(t)) = 0$$ where the potentialV isT-periodic in the time variable.     

Global uniqueness of homoclinic orbits for a class of fourth order equations

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation

Existence of even homoclinic orbits for second-order Hamiltonian systems

Some existence theorems for even homoclinic orbits are obtained for a class of second-order nonautonomous Hamiltonian systems with symmetric potentials under a class of new superquadratic conditions. A homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil-boundary-value problems which are obtained by the minimax methods.     

Infinitely many homoclinic orbits for the second-order Hamiltonian systems

In this paper, the existence of homoclinic orbits for the second-order Hamiltonian systems without periodicity is studied and infinitely many homoclinic orbits for both superlinear and asymptotically linear cases are obtained.     

一类二阶渐近周期微分方程的正同宿轨道

§ 1　IntroductionIn this note we are concerned with the asymptotically periodic second order equation-u″+α( x) u =β( x) uq +γ( x) up,　 x∈ R,( 1 )where1 相似文献