Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems
We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞<t<+∞, u=u1,u2, …,uN∈ℝNN≥3, a:ℝ→ℝ is a continuous bounded function, and the potential W:ℝN∖{ξ}→ℝ has a singularity at 0≠ξ∈ℝN, and...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/829052 |
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| author | Ziheng Zhang Fang-Fang Liao Patricia J. Y. Wong |
| author_facet | Ziheng Zhang Fang-Fang Liao Patricia J. Y. Wong |
| author_sort | Ziheng Zhang |
| collection | DOAJ |
| description | We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞<t<+∞, u=u1,u2, …,uN∈ℝNN≥3, a:ℝ→ℝ is a continuous bounded function, and the potential W:ℝN∖{ξ}→ℝ has a singularity at 0≠ξ∈ℝN, and
Wuu is the gradient of W at u. The novelty of this paper is that, for the case that N≥3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W. Different from the cases that (HS) is autonomous at≡1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N≥3. Besides the usual conditions on W, we need the assumption that a′t<0 for all t∈ℝ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved. |
| format | Article |
| id | doaj-art-126678798f6e491eb93fea12a6a176c9 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-126678798f6e491eb93fea12a6a176c92025-02-03T01:03:43ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/829052829052Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian SystemsZiheng Zhang0Fang-Fang Liao1Patricia J. Y. Wong2Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, ChinaDepartment of Mathematics, College of Science, Hohai University, Nanjing 210098, ChinaSchool of ELectrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, SingaporeWe are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞<t<+∞, u=u1,u2, …,uN∈ℝNN≥3, a:ℝ→ℝ is a continuous bounded function, and the potential W:ℝN∖{ξ}→ℝ has a singularity at 0≠ξ∈ℝN, and Wuu is the gradient of W at u. The novelty of this paper is that, for the case that N≥3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W. Different from the cases that (HS) is autonomous at≡1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N≥3. Besides the usual conditions on W, we need the assumption that a′t<0 for all t∈ℝ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.http://dx.doi.org/10.1155/2014/829052 |
| spellingShingle | Ziheng Zhang Fang-Fang Liao Patricia J. Y. Wong Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems Abstract and Applied Analysis |
| title | Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems |
| title_full | Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems |
| title_fullStr | Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems |
| title_full_unstemmed | Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems |
| title_short | Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems |
| title_sort | homoclinic solutions for a class of second order nonautonomous singular hamiltonian systems |
| url | http://dx.doi.org/10.1155/2014/829052 |
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