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Homoclinic solutions for a class of second-order Hamiltonian systems. (English) Zbl 1185.34056

This paper considers the second-order Hamiltonian systems

u ¨(t)+V(t,u(t))=f(t),( HS )

where t, u N , VC 1 (× N ,), and f: N . The authors prove the existence of a homoclinic solution of (HS) as the limit of 2KT-periodic solutions of

u ¨(t)=-V(t,u(t))+f k (t),( HS k )

where f k : N is a 2kT-periodic extension of f to the interval [-kT,kT),k· The main results are the following.

Theorem 1.1. Suppose that V and f0 satisfies the following conditions

V(t,x)=-K(t,x)+W(t,x) is T-periodic with respect to t,T>0

There exist constants b>0 and γ[1,2] such that

K(t,0)=0,K(t,x)b|x| γ forall(t,x)[0,T]× N ;

There exists a constant ϱ[2,μ] such that

(x,K(t,x))ϱK(t,x)forall(t,x)[0,T]× N ;

W(t,x)=o(|x|) as |x|0 uniformly with respect to t;

There is a constant μ>2 such that for all (t,x)×( N 0)

0<μW(t,x)(x,W(t,x));

f: N is a continuous and bounded function.

|f(t)| 2 dt<2(min{δ 2,bδ γ-1 -Mδ μ-1 }) 2 , where

M=sup{W(t,x)|t[0,T],x N ,|x|=1}

and δ(0,1] such that

bδ γ-1 -Mδ μ-1 =max x[0,1] (bx γ-1 -Mx μ-1 )·

Then system (HS) possesses a nontrivial homoclinic solution.

Theorem 1.2. Suppose that V and f=0 satisfies (H1), (H2 ' ), (H4)-(H6) and the following (H3 '' ) There exists a constant ϱ[2,μ] such that

K(t,x)(x,K(t,x))ϱK(t,x)for(t,x)[0,T]× N ;

Then system (HS) possesses a nontrivial homoclinic solution.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods