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.     

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.     

Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems

The existence of homoclinic solutions is obtained for second-order Hamiltonian systems , as the limit of the solutions of a sequence of nil-boundary-value problems which are obtained by the Mountain Pass theorem, when L(t) and W(t,x) are neither periodic nor even with respect to t.     

Existence of homoclinic solution for the second order Hamiltonian systems

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems , ∀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)/|x|²→+∞ as |x|→∞ uniformly in t, and need not satisfy the global Ambrosetti-Rabinowitz condition.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Homoclinic orbits for the second-order Hamiltonian systems with obstacle item

This paper is concerned with the existence of homoclinic orbits for the second-order Hamiltonian system with obstacle item, ü(t)-A u(t) =▽F (t, u), where F (t, u) is T-periodic in t with ▽F (t, u) = L(t)u + ▽R(t,u). By using a generalized linking theorem for strongly indefinite functionals, we prove the existence of homoclinic orbits for both the super-quadratic case and the asymptotically linear one.     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

Multiple homoclinic orbits for a class of the discrete p‐Laplacian with unbounded potentials

This paper is devoted to study the existence results of a sequence of infinitely many homoclinic orbits for the discrete p‐Laplacian with unbounded potentials without the Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem using the mountain pass theorem, the fountain theorem, and dual fountain theorem.