Abstract
In this paper, by using the mountain pass theorem, we investigate the existence of subharmonic weak solutions for a class of secondorder 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.
Keywords:
impulsive Lagrangian systems; damped term; subharmonic weak solutions; mountain pass theorem1 Introduction and main results
In this paper, we investigate the existence of subharmonic weak solutions for the following secondorder impulsive Lagrangian system with damped term:
where
(A)
for all
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 secondorder Hamiltonian systems are considered by many authors:
where
Theorem A (see [10], Theorem 1.1)
Assume thatFsatisfies
(F)
(H1) There exist constants
(H2)
(H3)
(H4) There exists a function
and
(H5) There exist constants
(H6) There exists
Then system (1.2) has a nontrivialTperiodic 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 [1520].
In [15], Nieto and O’Regan considered a onedimensional 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 [16], Nieto introduced a variational formulation for the following onedimensional 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:
where
For higher dimensional dynamical systems, some interesting results have also been obtained (see [2123]). In [21], Zhou and Li investigated the secondorder Hamiltonian system with impulsive effects:
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 [22], system (1.5) with
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:
where
In 2010, Li et al.[28] investigated the following system, more general than system (1.6), with
Motivated by [28], in [29], we investigated the following system, more general than system (1.7):
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 [29].
In [32], Luo et al. investigated the existence of subharmonic solutions with prescribed minimal period for the following onedimensional secondorder impulsive differential equation:
where
In this paper, motivated by [10,15,16,21,28,29] and [32], 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 kTperiodic for some
Let
and
In this paper, we make the following assumptions:
(P) There exists a constant
(K1) There exist constants
(K2)
(K3) There exists
(W1)
(W2) There exist constants
(W3) There exists a function
and
where
(W4) There exists
(W5) There exists a constant
(I1) There exist constants
(I2)
(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.
2 Preliminaries
In this section, we present the variational structure of system (1.1), which is motivated by [1517,28] and [29].
Let
Define
and
for each
is also a norm on
(see Proposition 1.1 in [1]). Hence, there exist positive constants
For any
If
Definition 2.1 Assume that
Remark 2.1 In [32], impulsive effects may occur periodically in
Note that
Then system (2.2) is equivalent to system (1.1) and its solutions are the solutions of system (1.1).
By the idea in [15], we take
by v and integrate it from 0 to kT. Then we obtain
Note that
Definition 2.2
holds for any
Lemma 2.1If
Proof Motivated by [15], for
Choose
Equations (2.5) and (2.6) imply that
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
It follows from assumption (A) and Theorem 1.4 in [1] that the functional
for
We will use the following mountain pass theorem to prove our results.
Lemma 2.2 (see [30])
LetEbe a real Banach space, and let
(i)
(ii) There exist constants
(iii) There exists
where
Remark 2.2 As shown in [31], 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
3 Main results
Theorem 3.1Assume that (P), (K1), (K2), (W1)(W4) and (I1)(I3) hold. Then, for every
Proof We use Lemma 2.2 to prove the theorem. Let
Step 1. We prove that
Choose
Step 2. We prove 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
Step 3. We prove that φ satisfies the (C)condition on
Then it follows from antisymmetry of B, (K2) and (I3) that
Next we prove that
Hence, we have
Let
Then it follows from
Let
It follows from (3.8) and Lemma 1 in [6] that there exists a subset
By (W3), we have
Let
which contradicts (3.5). Hence
Finally, (K1), (W1) and (I2) imply that
Remark 3.1 It is easy to see that Theorem 3.1 generalizes Theorem A. To be precise, when
Theorem 3.2Assume (P), (K1)(K3), (W1)(W5) and (I1)(I3) hold. Then system (1.1) has a sequence of distinct subharmonic weak solutions with period
Proof By Theorem 3.1, we know that for every
where
Let
Hence,
Obviously, we can find
Then, by (3.10), we have
4 Example
The following example is inspired partially by Example 3.1 in [10]. Let
where
Obviously, the condition (P) holds and
Then (W3) holds with
Hence, it is easy to see that there exists
So (I3) holds. Hence, by Theorem 3.1, we obtain that system (4.1) has at least one
kTperiodic solution for every
Moreover, it is easy to see that (K3) holds with
Choose
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
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).
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