Abstract
In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous secondorder Hamiltonian systems by using the variant fountain theorem.
1 Introduction
Consider the secondorder Hamiltonian systems
where
(A)
for all
Here and in the sequel,
There have been many investigations on the existence and multiplicity of periodic
solutions for Hamiltonian systems via the variational methods (see [17] and the references therein). In [6], Zhang and Liu studied the asymptotically quadratic case of
(AQ_{1})
(AQ_{2})
(AQ_{3})
They obtained the existence of infinitely many periodic solutions of (1.1) provided
The subquadratic condition (AQ_{1}) is widely used in the investigation of nonlinear differential equations. This condition
was weakened by some researchers; see, for example, [4] of Jiang and Tang. This paper considers the case of
(
The condition (
By the assumption (A) and the condition (
for
Meanwhile, we weaken the condition (AQ_{3}) to (
(
Then our main result is the following theorem.
Theorem 1.1Assume that (
Remark The conditions (AQ_{1}) and (AQ_{3}) are stronger than (
2 Preliminaries
In this section, we establish the variational setting for our problem and give the
variant fountain theorem. Let
We define the functional on E by
where
Define a selfadjoint linear operator
with the domain
and E possesses orthogonal decomposition
We can define on E a new inner product and the associated norm by
and
Therefore, Φ can be written as
Direct computation shows that
for all
Denote by
We state an abstract critical point theorem founded in [8]. Let E be a Banach space with the norm
Theorem 2.1 [[8], Theorem 2.2]
Assume that the functional
(T_{1})
(T_{2})
(T_{3}) There exist
and
Then there exist
Particularly, if
In order to apply this theorem to prove our main result, we define the functionals
A, B and
and
for all
3 Proof of Theorem 1.1
We firstly establish the following lemmas.
Lemma 3.1Assume that (
Proof Since
By the proof of Lemma 2.6 of [6], for any finitedimensional subspace
where
For the ϵ given in (3.1), let
Then
where
for any
for any
Lemma 3.2Assume that (
and
where
Proof Comparing this lemma with Lemma 2.7 of [6], we find that these two lemmas have the same condition (AQ_{2}) which is the key in the proof of Lemma 2.7 of [6]. We can prove our lemma by using the same method of [6], so the details are omitted. □
Now it is the time to prove our main result Theorem 1.1.
Proof of Theorem 1.1 By virtue of (1.3), (2.3) and (2.5),
For the sake of notational simplicity, in the following we always set
Step 1. We firstly prove that
Since
More precisely,
Now, we prove that
Put
By (1.3), we have
where
Passing to the limit in the inequality, by using
Thus,
By (1.2), we have
and by the assumption (A), we obtain
where
for all
An application of Fatou’s lemma yields
which is a contradiction to (3.8).
Step 2. We prove that
Since
for some
Note that
where
In view of the compactness of
Now, from the last assertion of Theorem 2.1, we know that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.
Acknowledgements
The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).
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