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2014 Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems
Ziheng Zhang, Fang-Fang Liao, Patricia J. Y. Wong
Abstr. Appl. Anal. 2014: 1-8 (2014). DOI: 10.1155/2014/829052

Abstract

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, ,uNNN3, a: is a continuous bounded function, and the potential W:N{ξ} has a singularity at 0ξN, 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 at1 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 at<0 for all t to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved.

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Ziheng Zhang. Fang-Fang Liao. Patricia J. Y. Wong. "Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems." Abstr. Appl. Anal. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/829052

Information

Published: 2014
First available in Project Euclid: 2 October 2014

zbMATH: 07023152
MathSciNet: MR3173293
Digital Object Identifier: 10.1155/2014/829052

Rights: Copyright © 2014 Hindawi

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