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Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term
Boundary Value Problems volume 2013, Article number: 218 (2013)
In this paper, by using the mountain pass theorem, we investigate the existence of subharmonic weak solutions for a class of second-order impulsive Lagrangian systems with damped term under asymptotically quadratic conditions. Some new existence criteria are established. Finally, an example is presented to verify our results.
MSC:37J45, 34C25, 70H05.
1 Introduction and main results
In this paper, we investigate the existence of subharmonic weak solutions for the following second-order impulsive Lagrangian system with damped term:
where , , , , , satisfying and , B is a skew-symmetric constant matrix, and are symmetric and continuous matrix-value functions on ℝ satisfying and , and satisfies , where K, W are T-periodic in their first variable, and the following assumption:
is measurable in t for every and continuously differentiable in x for a.e. , and there exist and with such that
for all and a.e. .
Lagrangian systems are applied extensively to study the fluid mechanics, nuclear physics and relativistic mechanics. Especially, as a special case of Lagrangian systems, the following second-order Hamiltonian systems are considered by many authors:
where satisfies , and the existence and multiplicity of periodic solutions, subharmonic solutions and homoclinic solutions are obtained via variational methods. We refer readers to [1–14]. Especially, in 2010, under the asymptotically quadratic conditions, Tang and Jiang  obtained the following interesting result.
Theorem A (see , Theorem 1.1)
Assume that F satisfies
and are T-periodic in their first variable with , and that K and W satisfy the following assumptions:
(H1) There exist constants and such that
(H2) for ;
(H3) uniformly for ;
(H4) There exists a function such that
(H5) There exist constants and such that
(H6) There exists such that
Then system (1.2) has a nontrivial T-periodic solution.
In recent years, variational methods have been applied to study the existence and multiplicity of solutions for impulsive differential equations and lots of interesting results have been obtained, see [15–20].
In , Nieto and O’Regan considered a one-dimensional Dirichlet boundary value problem with impulses. They obtained that the solutions of the impulsive problem minimize some (energy) functional and the critical points of the functional are indeed solutions of the impulsive problem.
In , Nieto introduced a variational formulation for the following one-dimensional damped nonlinear Dirichlet problem with impulses:
and gave the concept of a weak solution for such a problem. They obtained that the weak solutions of problem (1.3) are indeed the critical points of the functional:
By using the least action principle and the saddle point theorem, they obtained some existence results of solutions under sublinear condition and some reasonable conditions. In , system (1.5) with , where , was also investigated. By using variational methods, the authors obtained that system (1.5) has at least three weak solutions. In , the authors investigated system (1.5) with . They obtained that system (1.5) has infinitely many solutions under the assumptions that nonlinear term is superquadratic, asymptotically quadratic and subquadratic, respectively.
In recent years, via variational methods, some authors have been interested in studying the existence and multiplicity of periodic solutions and homoclinic solutions for the following Lagrangian systems with damped term:
In 2010, Li et al.  investigated the following system, more general than system (1.6), with :
By variational methods, under superquadratic or subquadratic conditions, we obtained that system (1.8) has infinitely many solutions. One can see more details of our results and more research background of system (1.8) in .
In , Luo et al. investigated the existence of subharmonic solutions with prescribed minimal period for the following one-dimensional second-order impulsive differential equation:
where , , , , , and if , while if .
In this paper, motivated by [10, 15, 16, 21, 28, 29] and , we focus on the existence of subharmonic weak solutions for system (1.1), which is of impulsive conditions, and we study the problem under asymptotically quadratic conditions. To the best of our knowledge, there are few papers that consider such a problem for system (1.1). We call a solution u subharmonic if u is kT-periodic for some .
In this paper, we make the following assumptions:
There exists a constant such that the matrix satisfies
(K1) There exist constants and such that
(K2) for all and a.e. ;
(K3) There exists such that
(W1) uniformly for a.e. ;
(W2) There exist constants and such that
(W3) There exists a function such that
(W4) There exists such that
(W5) There exists a constant such that
(I1) There exist constants () such that
(I2) and for all ;
(I3) There exists a constant C such that
This paper is organized as follows. In Section 2, we present the definition of a subharmonic classical solution, a subharmonic weak solution and the variational structure for system (1.1) and make some preliminaries. In Section 3, we present our main theorems and their proofs. In Section 4, an example is given to verify our main theorems.
for each . Then is a Hilbert space. It is well known that
is also a norm on . Obviously, if the condition (P) holds, and are equivalent. Moreover, there exists such that
(see Proposition 1.1 in ). Hence, there exist positive constants , such that
For any , define
If , then may not hold, which leads to impulsive effects.
Definition 2.1 Assume that and the limits and () exist. If u satisfies system (1.1), then we say that u is a subharmonic classical solution of system (1.1).
Remark 2.1 In , impulsive effects may occur periodically in , . In order to obtain a sequence of distinct subharmonic weak solutions (see Theorem 3.2 below), different from , in Definition 2.1, we assume that the impulsive effects only occur in , , which belong to . In other words, u is absolutely continuous on ℝ and is absolutely continuous on . Moreover, note that . Then it is easy to see that .
Note that . Then, by T-periodicity of q, we have . Moreover, obviously, is continuous on ℝ. We transform system (1.1) into the following system:
Then system (2.2) is equivalent to system (1.1) and its solutions are the solutions of system (1.1).
By the idea in , we take and multiply the two sides of the equality
by v and integrate it from 0 to kT. Then we obtain
Note that , is continuous on ℝ and . By integration by parts and the continuity of v, we obtain
Definition 2.2 is called a subharmonic weak solution of system (1.1) if
holds for any .
Lemma 2.1 If is a subharmonic weak solution of system (1.1), then u is a subharmonic classical solution of system (1.1).
Proof Motivated by , for , choose with for every . Then, by Definition 2.2, we obtain
Choose with for every . Then we obtain
Equations (2.5) and (2.6) imply that and u satisfies
Multiplying the above equality by v and integrating between 0 and kT, combining the argument of (2.4) and Definition 2.2, we obtain that
Hence, for every . This completes the proof. □
For every , define by
It follows from assumption (A) and Theorem 1.4 in  that the functional is continuously differentiable and
for . Obviously, if is a critical point of , i.e., , then is a subharmonic weak solution of system (1.1).
We will use the following mountain pass theorem to prove our results.
Lemma 2.2 (see )
Let E be a real Banach space, and let satisfy the (PS) condition. If ϕ satisfies the following conditions:
There exist constants such that ;
There exists such that , then ϕ possesses a critical value given by
where is an open ball in E of radius ρ centered at 0, and
Remark 2.2 As shown in , a deformation lemma can be proved by replacing the usual (PS)-condition with the condition (C), and it turns out that Lemma 2.2 holds true under the condition (C). We say that ϕ satisfies the condition (C), i.e., for every sequence , has a convergent subsequence if is bounded and as .
3 Main results
Theorem 3.1 Assume that (P), (K1), (K2), (W1)-(W4) and (I1)-(I3) hold. Then, for every , system (1.1) has at least one kT-periodic weak solution in .
Proof We use Lemma 2.2 to prove the theorem. Let .
Step 1. We prove that satisfies assumption (ii) of Lemma 2.2. It follows from (W1) and (W2) that there exist , and such that
Choose such that . Then it follows from (K1), (I2), (3.1) and (2.1) that for all with ,
Step 2. We prove that satisfies assumption (iii) of Lemma 2.2. Set for . By the argument in , we know that (W3) implies that
and (K2) implies that
It follows from (3.2), (3.3), (W3) and (I1) that for sufficiently large s,
By (W4), we can choose sufficiently large such that and . Let . Then satisfies assumption (iii) of Lemma 2.2.
Step 3. We prove that φ satisfies the (C)-condition on . The proof is motivated by . For every , assume that there exists a constant such that
Then it follows from antisymmetry of B, (K2) and (I3) that
Next we prove that is bounded. Assume that as . Let . Then , and so there exists a subsequence, still denoted by , such that on . Then, by Proposition 1.2 in , we get . Hence, we have and for a.e. . Thus, by conditions (P), (W2) and (I2), we have
Hence, we have
Let . Then, by (3.4), we get
Then it follows from and (3.6) that and so . Let and . Then and
Let . Then (3.7) and T-periodicity of in t imply that
It follows from (3.8) and Lemma 1 in  that there exists a subset of with such that
By (W3), we have
Let . Then by Fatou’s lemma and (3.9), we have
which contradicts (3.5). Hence is bounded. Going if necessary to a subsequence, assume that in . Then, by Proposition 1.2 in , we have and so as . Similar to the argument of Theorem 3.1 in , it is easy to obtain that . Hence, as . Hence, satisfies the (C)-condition.
Finally, (K1), (W1) and (I2) imply that . Hence, combining Step 1-Step 3 with Lemma 2.2 and Remark 2.2, we obtain that has at least a critical point in and . Then system (1.1) has at least one kT-periodic solution in . This completes the proof. □
Remark 3.1 It is easy to see that Theorem 3.1 generalizes Theorem A. To be precise, when , , , , and , Theorem 3.1 reduces to Theorem A.
Theorem 3.2 Assume (P), (K1)-(K3), (W1)-(W5) and (I1)-(I3) hold. Then system (1.1) has a sequence of distinct subharmonic weak solutions with period satisfying and as .
Proof By Theorem 3.1, we know that for every , system (1.1) has at least one kT-periodic solution in and . By Lemma 2.2, we have
Let , . Obviously, . Hence, by (K3), (W5) and (I1), we have
Hence, is uniformly bounded for all .
Obviously, we can find such that , then we claim that is distinct to for all . In fact, if for some , it is easy to check that
Then, by (3.10), we have , a contradiction. We can also find such that for all . Otherwise, if for some , we have . Then by (3.10), we have , a contradiction. In the same way, we can obtain that system (1.1) has a sequence of distinct periodic solutions with period satisfying and as . This completes the proof. □
The following example is inspired partially by Example 3.1 in . Let , . Consider the following impulsive Lagrangian system with damped term:
Obviously, the condition (P) holds and , and (K1), (K2), (W1) and (W2) hold with and .
Then (W3) holds with . Moreover,
Hence, it is easy to see that there exists such that (W4) holds by the above inequality. Obviously, (I1) and (I2) hold. Note that
So (I3) holds. Hence, by Theorem 3.1, we obtain that system (4.1) has at least one kT-periodic solution for every .
Moreover, it is easy to see that (K3) holds with . Since
Choose . Then (W5) holds. Hence, by Theorem 3.2, we obtain that system (4.1) has a sequence of distinct subharmonic weak solutions with period satisfying and as .
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This work is supported by Tianyuan Fund for Mathematics of the National Natural Science Foundation of China (No. 11226135) and the Fund for Fostering Talents in Kunming University of Science and Technology (No. KKSY201207032).
The author declares that he has no competing interests.
The author read and approved the final manuscript.