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.
Consider the second-order Hamiltonian systems
There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [1-7] and the references therein). In , Zhang and Liu studied the asymptotically quadratic case of under the following assumptions:
They obtained the existence of infinitely many periodic solutions of (1.1) provided is even inu (see Theorem 1.1 of ).
The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example,  of Jiang and Tang. This paper considers the case of , then . Motivated by  and , we replace (AQ1) with the following condition:
Then our main result is the following theorem.
Remark The conditions (AQ1) and (AQ3) are stronger than () and (). Then Theorem 1.1 above is different from Theorem 1.1 of .
We define the functional on E by
We can define on E a new inner product and the associated norm by
Therefore, Φ can be written as
Direct computation shows that
We state an abstract critical point theorem founded in . Let E be a Banach space with the norm and with for any . Set and . Consider the following -functional defined by
Theorem 2.1 [, Theorem 2.2]
3 Proof of Theorem 1.1
We firstly establish the following lemmas.
By the proof of Lemma 2.6 of , for any finite-dimensional subspace , there exists a constant such that
For the ϵ given in (3.1), let
Proof Comparing this lemma with Lemma 2.7 of , we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of . We can prove our lemma by using the same method of , 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), maps bounded sets to bounded sets uniformly for . Obviously, for all since is even in u. Consequently, the condition (T1) of Theorem 2.1 holds. Lemma 3.1 shows that the condition (T2) holds, whereas Lemma 3.2 implies that the condition (T3) holds for all , where is given there. Therefore, by Theorem 2.1, for each , there exist and such that
By (1.3), we have
By (1.2), we have
and by the assumption (A), we obtain
An application of Fatou’s lemma yields
which is a contradiction to (3.8).
The authors declare that they have no competing interests.
HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.
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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