Homoclinic orbits for a special class of non autonomous Hamiltonian systems

Salvatore, Addolorata

doi:10.1016/s0362-546x(97)00142-9

Cited by 49 publications

(26 citation statements)

References 10 publications

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“…An extension of the results in [39,40] can be found in [41], where a(t) fulfills some other types of coercive conditions. Clearly, the coercive conditions mentioned above are too restrictive, since they fails to cover the case that a(t) is bounded.…”

Section: Introductionmentioning

confidence: 76%

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Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

Zhang

Yuan

2015

J. Appl. Math. Comput.

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In this paper we study the existence of two almost homoclinic solutions for the following second order p(t)-Laplacian Hamiltonian systems with a small perturbation t, u) at u, f ∈ C(R, R n ) and belongs to L q(t) (R, R n ). The point is that, assuming that a(t) is bounded in the sense that there are two constants 0 < τ 1 < τ 2 < ∞ such that τ 1 ≤ a(t) ≤ τ 2 for all t ∈ R, W (t, u) is of super-p(t) growth as |u| → ∞ and satisfies some other reasonable hypothesis, f is sufficiently small in L q(t) (R, R n ), we provide one new criterion to ensure the existence of two almost homoclinic solutions. Recent results in the literature are extended and significantly improved.

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Section: Introductionmentioning

confidence: 76%

“…Under this coercive condition (A), Salvatore [39,40] introduced a compact embedding theorem, which has been utilizing from then on and plays a crucial role in demonstrating that the functional corresponding to systems (1.1) verifies the (PS) condition. Actually, in [39,40] Salvatore dealt with the following systems…”

Section: Introductionmentioning

confidence: 99%

Existence of two almost homoclinic solutions for p(t)-Laplacian Hamiltonian systems with a small perturbation

Zhang

Yuan

2015

J. Appl. Math. Comput.

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“…Later some papers weakened this condition [6,7]. There are also some other papers considered the sub-quadratic case [8,9] and the asymptotically quadratic case [10,11]. We all know that the main difficulty is to check the (PS) condition when one uses the Mountain Pass theorem.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Even homoclinic orbits for super quadratic Hamiltonian systems

Ding

Zhang

2010

Math. Meth. Appl. Sci.

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We study the existence of even homoclinic orbits for the second-order Hamiltonian systemü+V u (t, u) = 0. Let V(t, u) = −K(t, u)+W(t, u) ∈ C 1 (R×R n , R), where K is less quadratic and W is super quadratic in u at infinity. Since the system we considered is neither autonomous nor periodic, the (PS) condition is difficult to check when we use the Mountain Pass theorem. Therefore, we approximate the homoclinic orbits by virtue of the solutions of a sequence of nil-boundary-value problems.

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“…In [1,2], and [3], the authors considered homoclinic solutions for the special Hamiltonian system (1.3) in a weighted Sobolev space and obtained some results by using the mountain pass theorem in critical point theory. For the applications of mountain pass theorem, please see the references [4] and [5].…”

Section: U(t) -A(t) U(t)mentioning

confidence: 99%