#### Abstract

We establish a new existence result on homoclinic solutions for a second-order nonperiodic Hamiltonian systems. This homoclinic solution is obtained as a limit of solutions of a certain sequence of nil-boundary value problems which are obtained by the minimax methods. Some recent results in the literature are generalized and extended.

#### 1. Introduction

Consider the following second-order Hamiltonian system: where , , are maps. We will say that a solution of is homoclinic (to ), if , as . In addition, if , then is called a nontrivial homoclinic solution.

Inspired by the excellent monographs [1, 2], by now, the existence and multiplicity of homoclinic solutions for Hamiltonian systems have been extensively investigated in many papers via variational methods; see [3–7] for the first order systems and [8–19] for the second systems, and most of them treat the following system: where is a symmetric matrix-valued function and .

For the periodic case, the periodicity is used to control the lack of compactness due to the fact that (1) is set on all . In 1990, Rabinowitz [12] first proved that (1) has a -periodic solution , which is bounded uniformly for , and obtained a homoclinic solution for (1) as a limit of -periodic solution.

For the nonperiodic case, the problem is quite different from the one described in nature. Rabinowitz and Tanaka [13] introduced a type of coercive condition on the matrix : , as .

They first obtained the existence of homoclinic solution for the nonperiodic system (1) under the well-known (AR) growth condition by using Ekeland’s variational principle.

In 1995, Ding [8] strengthened condition by there exists a constant such that Under the condition and some subquadratic conditions on , Ding proved the existence and multiplicity of homoclinic solutions for the system (1). From then on, the condition or is extensively used in nonperiodic second-order Hamiltonian systems. However, the assumption or is a rather restrictive and not very natural condition as it excludes, for example, the case of constant matrices .

In 2005, Izydorek and Janczewska [9] first presented the “pinching” condition (see the following ()) and relaxed the conditions and . They studied the general periodic Hamiltonian system where and obtained the following result.

Theorem A (see [9]). *Let the following conditions hold: * *, where is continuous and periodic with respect to , ;* *there exist such that
* * for all ;* * as uniformly in ;* *there is a constant such that
* * and , where and is a positive constant depending on . ** Then the system (3) possesses a nontrivial homoclinic solution such that as .*

From then on, following the idea of [9], some researchers are devoted to relaxing the conditions and and studying the existence of homoclinic solutions of system or (3) under the periodicity assumption of the potential, such as [10, 11, 16, 19].

Very recently, Daouas [3] removed the periodicity condition and studied the existence of homoclinic solutions for the nonperiodic system (3), when is superquadratic at the infinity. Motivated by [3], in this work, we will study the existence of homoclinic solutions of the nonperiodic system , when satisfies the asymptotically quadratic condition at the infinity. It is worth noticing that there are few works concerning this case for system or (3) up to now.

Our result is presented as follows.

Theorem 1. *Let hold. Moreover, assume that the following conditions hold:* *, and there exists a constant such that
* *there exists such that
* * and as uniformly in , and there exist, such that
for any and ;* * as uniformly in , where with ;* * as , and
for any . ** Then the system possesses a nontrivial homoclinic solution such that as .*

*Remark 2. *Theorem 1 treats the asymptotically quadratic case on . Consider the functions
where and .

A straightforward computation shows that and satisfy the assumptions of Theorem 1, but does not satisfy the conditions and . Hence, Theorem 1 also extends the results in [8, 13].

The remainder of this paper is organized as follows. In Section 2, some preliminary results are presented. In Section 3, we give the proof of Theorem 1.

#### 2. Preliminaries

Following the similar idea of [20], consider the following nil-boundary value problems: For each , let , where equipped with the norm Furthermore, for , let and under their habitual norms. We need the following result.

Proposition 3 (see [9]). *There is a positive constant such that for each and the following inequality holds:
*

Note that the inequality (14) holds true with constant if (see [9]). Subsequently, we may assume this condition is fulfilled.

Consider a functional defined by Then , and it is easy to show that for all , we have It is well known that critical points of are classical solutions of the problem (11). We will obtain a critical point of by using an improved version of the Mountain Pass Theorem. For completeness, we give this theorem.

Recall that a sequence is a -sequence for the functional if is bounded and . A functional satisfies the -condition if and only if any -sequence for contains a convergent subsequence.

Theorem 4 (see [21]). *Let be a real Banach space, and let satisfy the (C)-condition and . If satisfies the following conditions:* *there exist constants such that ;* *there exists such that , then possesses a critical value given by
* *where is an open ball in of radius at about , and
*

*Proof. *As shown in Bartolo et al. [22], a deformation lemma can be proved with the -condition replacing the usual -condition, and it turns out that the standard version Mountain Pass Theorem (see Rabinowitz [21]) holds true under the -condition.

Lemma 5. *Assume that holds, then
*

*Proof. *From it follows that for a map given by
is nondecreasing. Similar to the proof in [12], we can get the conclusion.

Lemma 6 (see [9]). *Let be a continuous map such that is locally square integrable. Then, for all , one has
*

#### 3. Proof of Theorem 1

Lemma 7. *Under the assumptions of Theorem 1, the problem (11) possesses a nontrivial solution.*

*Proof. *It suffices to prove that the functional satisfies all the assumptions of Theorem 4.*Step *1. We show that the functional satisfies the -condition. Let

Observe that, for large, it follows from () and () that there exists a constant such that
Arguing indirectly, assume as a contradiction that . Setting , then , and by Proposition 3, one has
Note that
where . This implies that
Set for
By , as .

For , let
Then by (), . One has
It follows from (23) that
which implies that
as uniformly in , and for any fixed
as . Using (14) and (31), we have
as uniformly in .

Let . By there is such that
for all . Consequently,
for all .

By (31), we can take large such that
Hence, by one has
for all . By (32) there exists such that

for all . By (35)–(38), one has
which contradicts with (26). So is bounded in . In a similar way to Proposition B. 35 in [21], we can prove that has a convergent subsequence. Hence satisfies the -condition.*Step *2. We show that the functional satisfies the condition of Theorem 4.

Observe that, by () and (), given , there exists some such that
for all and , where . It follows from (), (40), and Proposition 3 that
Hence there exist and such that for all with .*Step *3. We show that the functional satisfies the condition of Theorem 4.

By , there exists such that
Let
where and . Clearly, . By (15), (42), and Lemma 5, one has
Since and , then . So as . So, we can choose large enough such that and .

Clearly ; then, by application of Theorem 4, there exists a critical point of such that for all .

Lemma 8. * is bounded uniformly in .*

*Proof. *Define the set of paths
It follows from Lemma 7 that there exists a solution of problem (11) at which
is achieved. Let . Since any function in can be regarded as belonging to if one extends it by zero in , then . Therefore, for any solution of problem (11), we obtain
Notice that , and together with (47), one has
The rest of the proof is similar to that of Step 1 in Lemma 7. Hence there exists a constant , independent of such that
The proof is complete.

Take a sequence , and consider the problem (11) on the interval . By Lemma 7, there exists a nontrivial solution of problem (11).

Lemma 9. *Let be the sequence given above. Then there exists a subsequence convergent to in .*

*Proof. *First we prove that the sequences , , and are bounded. From (14) and (49), for large enough, one has
By (11) and (50), for all , there exists independent of such that
It follows from the Mean Value Theorem that for every and , there exists such that
Combining the above with (50), and (51) we get
and hence for large enough
Second we show that the sequences and are equicontinuous. Indeed, for any and , by (54), we have
Similarly, by (51), one gets
By using the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence and a function such that
The proof is complete.

Lemma 10. *Let be the function given by (57). Then is the homoclinic solution of .*

*Proof. *First we show that is a solution of . Let be the sequence given by Lemma 9, then
for every and . Take with . There exists such that for all ; we get and
Integrating (59) from to , we have
Since uniformly on and uniformly on as , then, from (60), we obtain
Because of the arbitrariness of and , we conclude that satisfies .

Second we prove that , as . Note that, by (49), for , there exists such that, for all , one has
Letting , one gets
and letting , we have
and so
From Lemma 6 and (65), we obtain as .

Next we show that as . Indeed, applying again Lemma 6 to , we obtain

Also, from (65), we get
Hence, it is enough to prove that
Since is a solution of , one has
Since for all and , as , (68) follows from (69).

Finally, similar to the proof in [12], we can prove that is nontrivial, and we omit it here. The proof of Theorem 1 is complete.