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Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition
Advances in Difference Equations volume 2014, Article number: 244 (2014)
By means of critical point theory and some analysis methods, the existence of homoclinic solutions for the p-Laplacian system with delay, , is investigated. Some new results are obtained. The interesting thing is that the function is only required to satisfy a local condition. Furthermore, the results are all explicitly related to the value of delay τ.
MSC:34C37, 58E05, 70H05.
In the past years, the existence of homoclinic solutions to some second-order ordinary differential systems has been extensively investigated because of its background in applied science (see [1–7] and the references cited therein). For example, in , XH Tan and Li Xiao studied the existence of homoclinic solutions to the p-Laplacian system
where is a constant. The following theorem was obtained.
Theorem 1.1 (See )
Assume that F and f satisfy the following conditions:
(B1) is T-periodic with respect to t, is a constant;
(B2) There are constants and such that for all
(B3) is a continuous and bounded function such that .
Then (1.1) possesses a homoclinic solution.
From the proof of Theorem 1.1 in , we see that assumption (B2) is crucial for obtaining the existence of a homoclinic solution for (1.1).
However, few papers investigated the existence of homoclinic solutions to functional differential equations [8–10]. In , the authors studied the existence of homoclinic solutions to the following functional differential equation:
where is a constant, , , and being τ-periodic in t. Under
[H1] there is a τ-periodic continuously differentiable function such that
[H2] there is a with such that
and some other conditions, they obtained the result that (1.2) possesses a nontrivial homoclinic orbit.
In this paper, we investigate further the existence of homoclinic solutions for a second-order p-Laplacian functional differential system as follows:
where , , and is T-periodic in t, , , , and are constants. The interesting thing of this paper is that the period T of function with respect to the variable t may not be equal to τ, and the main results are all expressively related to the value of delay τ. Furthermore, even if for the case of , we do not require condition (B2) for guaranteeing the coercive potential condition.
As is well known, a solution of (1.3) is named homoclinic (to 0) if and as . In addition, if , then u is called a nontrivial homoclinic solution.
Motivated by the idea in the work of PH Rabinowitz in , Marek Izydorek, Joanna Janczewska in  and XH Tan, Li Xiao in , the existence of a homoclinic solution for (1.5) is obtained as a limit of a certain sequence of -periodic solutions for the following equation:
where , is a -periodic extension of restriction of e to the interval . We obtain the following theorem.
Theorem 1.2 Assume that G and e satisfy the following conditions:
(C1) for all ;
(C2) there are constants , (), , and such that
where ρ is a constant with ;
(C3) is a continuous and bounded function such that .
Then (1.3) possesses a nontrivial homoclinic solution, if
In particular, suppose
where , , , and , , are all constants. If there is a constant such that
then from (1.6), one can easily find
and from (1.7), we have
So by using Theorem 1.2, we have the following result.
Corollary 1.1 Assume that assumption (C1), assumption (C3), condition (1.6) (or condition (1.7)) and condition (1.8) hold. Then (1.3) possesses a nontrivial homoclinic solution, if
Remark 1.1 If , and assumptions (C1)-(C3) are satisfied, then for sufficiently small and sufficiently large , we see that condition (1.5) is satisfied. So by using Theorem 1.2, we see that (1.3) possesses a nontrivial homoclinic solution. Furthermore, if , then (1.3) is converted to
where , and from (1.6) or (1.7), we see that the function is allowed to be
which implies condition (B2) for guaranteeing the coercive potential does not hold for (1.3). Moreover, we do not require that is τ-periodic function with respect to t, which is required by , and local condition (C2) is essentially different from assumption [H2] in .
If μ in assumption (C2) is the case , then we have further the following result.
Theorem 1.3 Assume that assumption (C1) in Theorem 1.2 is satisfied together with the following conditions:
(D2) there are constants , , (), and such that for all with , ,
(D3) is a continuous and bounded function such that .
Then (1.3) possesses a nontrivial homoclinic solution, if
Suppose there are constants , (), , (), , and such that for all ,
If there is a constant such that
then for all with , ,
So by using Theorem 1.3, we have the following result.
Corollary 1.2 Assume that assumption (C1) in Theorem 1.2, assumption (D3) in Theorem 1.3, condition (1.10) (or condition (1.11)) and condition (1.12) hold. Then (1.3) possesses a nontrivial homoclinic solution, if
where is determined in (1.12).
Throughout this paper, denotes the standard inner product in and is the induced norm. For each , denotes the Banach space of -periodic functions on R with values in under the norm
denotes the Banach space of -periodic functions on R with values in under the norm
and denotes the Banach space of -periodic essentially bounded measurable functions from R to with the norm
Let with and , where , , and σ are positive constants. Then as .
Lemma 2.2 
Let , and be constants, with or . Then for each with , we have
Lemma 2.3 
If is continuous differential on R, , and are constants, then for every the following inequality holds:
Lemma 2.4 
Let X be a real reflexive Banach space and be a bounded convex closed subset of X. Suppose that is a lower weakly semi-continuous functional. If there exists a point such that
then there must be a such that
In order to investigate the existence of homoclinic solutions to (1.3), we should study the existence of -periodic solutions to (1.4) for each in the first case.
Lemma 2.5 Assume that the functions G and e satisfy conditions (C1)-(C3), and also condition (1.5) holds. Then for each , (1.4) possesses a -periodic solution such that
where and are constants independent of k.
Proof For each , let be defined by
Then and is weakly lower semi-continuous. Furthermore, one can easily check
Since , it follows from (2.2) that
So if is a critical point of , then must be a -periodic solution to (1.6). Thus, we should prove that possesses a critical point. In order to do it, let and , where
is a constant defined by assumption (C2). From , we see Ω is a closed bounded convex subset of . Now, for using Lemma 2.4, we should prove that for each ,
If , then . So by using Lemma 2.3, we have
Substituting in (2.3) into the above formula,
So, for all , by using conditions (C1) and (C2),
By using the Hölder inequality and Lemma 2.2,
In view of condition (C3), we see
Substituting (2.5), (2.6), (2.7), and (2.8) into (2.4), and using the Young inequality,
By using the inequality (see )
where is a constant, we obtain from (2.9) the result that
which together with (1.5) yields
Thus by using Lemma 2.4, we see that for each , there is a point
In view of being an open subset of , we see from Theorem 1.3 in  that
and from (2.3) and the fact , we see
The proof is complete. □
Lemma 2.6 Assume that assumption (C1) of Theorem 1.2, assumptions (D2)-(D3) of Theorem 1.3 and condition (1.9) hold. Then for each , (1.4) possesses a -periodic solution such that
where is a constant determined by (D2) and (1.9). Clearly, ρ is independent of k.
Proof Let . Clearly, Γ is a bounded closed convex subset of . Similar to the proof of Lemma 2.5, it suffices to show that for each ,
If , then . So by using Lemma 2.3, we have
Furthermore, for all , by using assumptions (D2) and (D3), and arguing in a similar way to the proof of Lemma 2.5, we have
which together with (1.9) yields
The proof is complete. □
Lemma 2.7 
Let be the -periodic solution to (1.4) that satisfies (2.1) for each . Then there exists a subsequence of convergent to a in .
3 Proof of main result
Proof of Theorem 1.2 Firstly, we will prove that , which is determined by Lemma 2.7, is a solution to (1.3). Since is a -periodic solution to (1.4), it follows that
Take with , then there must be a positive integer such that for , . So for , for all , and then by (3.1)
Thus, by using Lemma 2.7, uniformly for , where
Since for and is continuous differential of on for every , it follows that on . In view of being arbitrary with , , , that is, , is a solution to (1.5).
Below, we will prove and as .
clearly, for every if , by (2.1),
Let and , respectively, we have
So by using Lemma 2.3,
Thus, , which, together with the fact that is a solution of (1.3), i.e.,
Furthermore, from (3.2), ; and then by using Lemma 2.1, we have
Combining (3.3) and (3.4), we see is a homoclinic solution to (1.3). Clearly, , otherwise, by substituting into (1.3), we have
By using assumption (C1), we get for all . So it follows from (3.5) that , which contradicts the fact that in assumption (C3). The proof is complete. □
Since the proof of Theorem 1.3 works almost exactly as the proof of Theorem 1.2, we omit the proof of Theorem 1.3 here.
For example, consider the following equation:
where , , and is a constant.
By calculating, we can choose such that (3.6) is rewritten as
Corresponding to Theorem 1.2, we see , . So we can choose , , , , , and such that conditions (C1) and (C2) are satisfied; and also
which implies that condition (C3) holds. So we can choose such that , and if is sufficiently small, then
Thus, by using Corollary 1.1, we see that (3.6) has a nontrivial homoclinic solution for small enough.
Especially, if , then (3.6) is converted to
So we can choose such that (3.7) is written as
Clearly, we can choose such that all the conditions of Corollary 1.1 are satisfied. So (3.7) has a nontrivial homoclinic solution.
Remark 3.1 From (3.7), we see that if set , then (1.1) is the special case of (3.6) for . Also, from (3.7), we see that
which implies that the crucial assumption (B2) for guaranteeing the coercive condition in  (see Theorem 1.1 in Section 1) does not hold. So the results in present paper are essentially new.
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Research supported by the NSF of China (No. 11271197).
The authors declare that they have no competing interests.
All authors have equally contributed to obtaining new results in this article and also read and approved the final manuscript.
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Lu, S., Lu, M. Homoclinic solutions for a second-order p-Laplacian functional differential system with local condition. Adv Differ Equ 2014, 244 (2014). https://doi.org/10.1186/1687-1847-2014-244
- critical point theory
- homoclinic solution
- periodic solution
- functional differential system