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Title: Homoclinic solutions for a class of second order nonautonomous singular Hamiltonian systems
Authors: Zhang, Ziheng
Liao, Fang-Fang
Wong, Patricia J. Y.
Keywords: DRNTU::Engineering::Electrical and electronic engineering
Issue Date: 2014
Source: Zhang, Z., Liao, F.-F., & Wong, P. J. Y. (2014). Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems. Abstract and Applied Analysis, 2014, 829052-.
Series/Report no.: Abstract and applied analysis
Abstract: We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems ̈ 𝑢??? + 𝑎??? ( 𝑡??? ) 𝑊??? 𝑢??? ( 𝑢??? ) = 0 , (HS) where − ∞ < 𝑡??? < + ∞ , 𝑢??? = ( 𝑢??? 1 , 𝑢??? 2 , … , 𝑢??? 𝑁??? ) ∈ ℝ 𝑁??? ( 𝑁??? ≥ 3 ) , 𝑎??? ∶ ℝ → ℝ is a continuous bounded function, and the potential 𝑊??? ∶ ℝ 𝑁??? \ { 𝜉??? } → ℝ has a singularity at 0 ≠ 𝜉??? ∈ ℝ 𝑁??? , and 𝑊??? 𝑢??? ( 𝑢??? ) 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 ( 𝑎??? ( 𝑡??? ) ≡ 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.
ISSN: 1085-3375
DOI: 10.1155/2014/829052
Schools: School of Electrical and Electronic Engineering 
Rights: © 2014 Ziheng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:EEE Journal Articles

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