Abstract
By applying the mountain pass theorem and the symmetric mountain pass theorem in critical point theory, the existence and multiplicity of fast homoclinic solutions are obtained for the following second-order non-autonomous problem: , where , , , , are not periodic in t and is a continuous function and with . MSC:34C37, 35A15, 37J45, 47J30.
Highlights
In order to introduce the concept of fast homoclinic solutions for problem ( . ), we first state some properties of the weighted Sobolev space E on which the certain variational functional associated with ( . ) is defined and the fast homoclinic solutions are the critical points of the certain functional
It follows that φ(u) < for u ∈ E and u ≥ dr , which shows that (iii) of Lemma
Summary
Consider fast homoclinic solutions of the following problem:. where p ≥ , t ∈ R, u ∈ RN , a ∈ C(R, R), W ∈ C (R × RN , R) are not periodic in t, and q : R → R is a continuous function and Q(t) =. Consider fast homoclinic solutions of the following problem:. ) reduces to the following special second-order Hamiltonian system:. ) reduces to the following second-order damped vibration problem:. There is little research as regards the existence of homoclinic solutions for damped vibration problems In , Wu and Zhou [ ] obtained some results for damped vibration problems Zhang [ ] obtained infinitely many solutions for a class of general second-order damped vibration systems by using the variational methods. Zhang [ ] investigated subharmonic solutions for a class of second-order impulsive systems with damped term by using the mountain pass theorem. In order to introduce the concept of fast homoclinic solutions for problem ) is defined and the fast homoclinic solutions are the critical points of the certain functional.
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