Abstract
In this paper, we deal with the existence of infinitely many homoclinic solutions for a class of second-order Hamiltonian systems. By using the dual fountain theorem, we give some new criteria to guarantee that the second-order Hamiltonian systems have infinitely many homoclinic solutions. Some recent results are generalised and significantly improved.
Highlights
Consider the following second-order Hamiltonian systems:u (t) – L(t)u(t) + Wu t, u(t) =, ∀t ∈ R, ( . )where L ∈ C(R, RN×N ) is a symmetric matrix valued function, u ∈ RN and W ∈ C (R × RN, R)
Inspired by the excellent monographs [, ], the existence of periodic solutions and homoclinic solutions for second-order Hamiltonian systems have been intensively studied in many recent papers via variational methods; see [ – ] and references therein
Some researchers have begun to study the existence of solutions for second-order Hamiltonian systems with impulses by using some critical points theorems of [, ]; see [, ]
Summary
Inspired by the excellent monographs [ , ], the existence of periodic solutions and homoclinic solutions for second-order Hamiltonian systems have been intensively studied in many recent papers via variational methods; see [ – ] and references therein. Some researchers have begun to study the existence of solutions for second-order Hamiltonian systems with impulses by using some critical points theorems of [ , ]; see [ , ]. ) has infinitely many homoclinic solutions by using the following conditions. There are many functions W satisfying our Theorem . It is easy to see that there are many functions W satisfying the conditions of Theorem . But not satisfying (A ), (A ), the condition (C ) in Theorem .
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