zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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)


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.

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