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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.
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  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  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  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  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 ). 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 , 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  removed the periodicity condition and studied the existence of homoclinic solutions for the nonperiodic system (3), when is superquadratic at the infinity. Motivated by , 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].
Following the similar idea of , 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 ). There is a positive constant such that for each and the following inequality holds:
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 ). 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. , 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 ) 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 , we can get the conclusion.
Lemma 6 (see ). Let be a continuous map such that is locally square integrable. Then, for all , 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
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 , 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.
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 , we can prove that is nontrivial, and we omit it here. The proof of Theorem 1 is complete.
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