Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

Gu Hua, An, An Tianqing

Description

In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem. In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

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• Publié par: biomed
• Publié le: 01 janvier 2013
• Nombre de lectures: 6

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Gu and AnBoundary Value Problems2013,2013:16 http://www.boundaryvalueproblems.com/content/2013/1/16

R E S E A R C H Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

* Hua Guand Tianqing An

* Correspondence: [e-mail] College of Science, Hohai University, Nanjing, 210098, China

Open Access

Abstract In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

1 Introduction Consider the second-order Hamiltonian systems   ¨ u(t) +∇ u W(t,u) = ,∀t∈R,  u() =u(T), ˙ u() =u ˙ (T),T> ,

(.)

whereW(t,u) is alsoT-periodic and satisfies the following assumption (A): N (A)W(t,u)is measurable intfor allu∈R, continuously differentiable inufor a.e. + + + t∈[,T]and there exista∈C(R,R)andb∈L([,T],R)such that       W(t,u)≤a|u|b(t),∇ u W(t,u)≤a|u|b(t)

N for allu∈Rand a.e.t∈[,T]. Here and in the sequel,∙,∙and| ∙ |always denote the standard inner product and the N norm inRrespectively. There have been many investigations on the existence and multiplicity of periodic so- lutions for Hamiltonian systems via the variational methods (see [–] and the references therein). In [], Zhang and Liu studied the asymptotically quadratic case ofW(t,u) =  U(t)u,u+W  (t,u) under the following assumptions:  N (AQ  )W  (t,u)≥for all(t,u)∈[,T]×R, and there exist constantsµ∈(, )andR  >  such that   ∇ u W  (t,u),u≤µW  (t,u),∀t∈[,T]and|u| ≥R  ;

W(t,u)  (AQ  )lim |u|→  =∞uniformly fort∈[,T], and there exist constantsc  ,R  > such |u| that

W  (t,u)≤c  |u|,

∀t∈[,T]and|u| ≤R  ;

©2013 Gu and An; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.