Homoclinic Solutions for a Second-Order Nonperiodic Asymptotically Linear Hamiltonian Systems

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.


Introduction
Consider the following second-order Hamiltonian system: : R × R → R are 1 maps. We will say that a solution : R → R of (HS) is homoclinic (to 0), if ( ) → 0, as | | → ∞. In addition, if ̸ ≡ 0, then is called a nontrivial homoclinic solution.
For the periodic case, the periodicity is used to control the lack of compactness due to the fact that (1) is set on all R. In 1990, Rabinowitz [12] first proved that (1) has a 2 -periodic solution , which is bounded uniformly for , and obtained a homoclinic solution for (1) as a limit of 2 -periodic solution.
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 (L 1 ) by (L 2 ) there exists a constant > 0 such that Under the condition (L 2 ) and some subquadratic conditions on ( , ), Ding proved the existence and multiplicity of homoclinic solutions for the system (1). From then on, the condition (L 1 ) or (L 2 ) is extensively used in nonperiodic second-order Hamiltonian systems. However, the assumption (L 1 ) or (L 2 ) is a rather restrictive and not very natural condition as it excludes, for example, the case of constant matrices .
Theorem A (see [9]). Let the following conditions hold: where is continuous and periodic with respect to , > 0; (V 2 ) there exist 1 , 2 > 0 such that From then on, following the idea of [9], some researchers are devoted to relaxing the conditions (L 1 ) and (L 2 ) and studying the existence of homoclinic solutions of system (HS) 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 (HS), when satisfies the asymptotically quadratic condition at the infinity. It is worth noticing that there are few works concerning this case for system (HS) or (3) up to now.
Our result is presented as follows.
A straightforward computation shows that and satisfy the assumptions of Theorem 1, but does not satisfy the conditions (L 1 ) and (L 2 ). 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.

Preliminaries
Following the similar idea of [20], consider the following nilboundary value problems: For each > 0, let → R is an absolutely continuous function, equipped with the norm Furthermore, for , R ) under their habitual norms. We need the following result.
Proposition 3 (see [9]). There is a positive constant such that for each > 0 and ∈ the following inequality holds: Abstract and Applied Analysis 3 Note that the inequality (14) holds true with constant = √ 2 if ≥ 1/2 (see [9]). Subsequently, we may assume this condition is fulfilled.
Consider a functional : → R defined by Then ∈ 1 ( , R), 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. Theorem 4 (see [21]). Let be a real Banach space, and let ∈ 1 ( , R) satisfy the (C)-condition and (0) = 0. If satisfies the following conditions: (A 2 ) there exists ∈ \ (0) such that ( ) ≤ 0, then possesses a critical value ≥ given by where ( 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 ( 2 ) holds, then Proof. From (H 2 ) it follows that for ̸ = 0 a map given by is nondecreasing. Similar to the proof in [12], we can get the conclusion.
Lemma 6 (see [9]). Let : R → R be a continuous map such thaṫis locally square integrable. Then, for all ∈ R, one has 3. Proof of Theorem 1 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 Arguing indirectly, assume as a contradiction that ‖ ‖ → ∞. Setting = /‖ ‖, then ‖ ‖ = 1, and by Proposition 3, one has Note that  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.
Proof. Define the set of paths It follows from Lemma 7 that there exists a solution of problem (11) is achieved. Let > . Since any function in can be regarded as belonging to if one extends it by zero in [− , ] \ [− , ], then Γ ⊂ Γ . Therefore, for any solution of problem (11), we obtain Notice that ( ) = 0, 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 1 > 0, independent of such that The proof is complete.
Take a sequence → ∞, and consider the problem 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 3 > 0 independent of such thaẗ It follows from the Mean Value Theorem that for every ∈ and ∈ R, there exists ∈ [ − 1, ] such thaṫ Combining the above with (50), and (51) we geṫ and hence for large enougḣ Second we show that the sequences { } ∈N and {̇} ∈N are equicontinuous. Indeed, for any ∈ N and 1 , 2 ∈ R, by (54), we have Abstract and Applied Analysis Similarly, by (51), one getṡ By using the Arzelà-Ascoli Theorem, we obtain the existence of a subsequence { } ∈N and a function 0 such that The proof is complete.
Lemma 10. Let 0 : R → R be the function given by (57). Then 0 is the homoclinic solution of (HS).
Proof. First we show that 0 is a solution of (HS). Let { } ∈N be the sequence given by Lemma 9, then  Because of the arbitrariness of and , we conclude that 0 satisfies (HS).