Subharmonic solutions for a class of second-order impulsive Lagrangian systems with damped term
© Zhang; licensee Springer. 2013
Received: 16 July 2013
Accepted: 27 August 2013
Published: 7 November 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
- (A)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. .
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)
and are T-periodic in their first variable with , and that K and W satisfy the following assumptions:
(H2) for ;
(H3) uniformly for ;
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.
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.
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 .
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 .
- (P)There exists a constant such that the matrix satisfies
(K2) for all and a.e. ;
(W1) uniformly for a.e. ;
(I2) and for all ;
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.
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 .
Then system (2.2) is equivalent to system (1.1) and its solutions are the solutions of system (1.1).
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).
Hence, for every . This completes the proof. □
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 )
There exist constants such that ;
- (iii)There exists such that , then ϕ possesses a critical value given by
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 .
By (W4), we can choose sufficiently large such that and . Let . Then satisfies assumption (iii) of Lemma 2.2.
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 .
Hence, is uniformly bounded for all .
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. □
So (I3) holds. Hence, by Theorem 3.1, we obtain that system (4.1) has at least one kT-periodic solution for every .
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 .
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).
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View ArticleMATHGoogle Scholar
- Long YM: Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials. Nonlinear Anal. TMA 1995, 24: 1665-1671. 10.1016/0362-546X(94)00227-9View ArticleMATHMathSciNetGoogle Scholar
- Ding YH: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 1995, 25(11):1095-1113. 10.1016/0362-546X(94)00229-BMathSciNetView ArticleMATHGoogle Scholar
- Schechter M: Periodic non-autonomous second-order dynamical systems. J. Differ. Equ. 2006, 223: 290-302. 10.1016/j.jde.2005.02.022MathSciNetView ArticleMATHGoogle Scholar
- Tang CL: Periodic solutions of nonautonomous second order systems with γ -quasisubadditive potential. J. Math. Anal. Appl. 1995, 189: 671-675. 10.1006/jmaa.1995.1044MathSciNetView ArticleMATHGoogle Scholar
- Tang CL, Wu XP: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 2001, 259: 386-397. 10.1006/jmaa.2000.7401MathSciNetView ArticleMATHGoogle Scholar
- Jiang Q, Tang CL: Periodic ad subharmonic solutions of a class of subquadratic second-order Hamiltonian systems. J. Math. Anal. Appl. 2007, 328: 380-389. 10.1016/j.jmaa.2006.05.064MathSciNetView ArticleMATHGoogle Scholar
- Wu X: Saddle point characterization and multiplicity of periodic solutions of non-autonomous second-order systems. Nonlinear Anal. 2004, 58: 899-907. 10.1016/j.na.2004.05.020MathSciNetView ArticleMATHGoogle Scholar
- Zhao F, Wu X: Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems. J. Math. Anal. Appl. 2005, 308: 588-595. 10.1016/j.jmaa.2004.11.046MathSciNetView ArticleMATHGoogle Scholar
- Tang XH, Jiang J: Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems. Comput. Math. Appl. 2010, 59: 3646-3655. 10.1016/j.camwa.2010.03.039MathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Zhou Y: Periodic solutions of non-autonomous second order Hamiltonian systems. J. Math. Anal. Appl. 2008, 345: 929-933. 10.1016/j.jmaa.2008.05.026MathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Tang X: Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems. Nonlinear Anal., Real World Appl. 2012, 13: 113-130. 10.1016/j.nonrwa.2011.07.013MathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Tang X: Existence of subharmonic solutions for non-quadratic second order Hamiltonian systems. Bound. Value Probl. 2013. 10.1186/1687-2770-2013-139Google Scholar
- Zhang Q, Liu C: Infinitely many homoclinic solutions for second order Hamiltonian systems. Nonlinear Anal. 2010, 72: 894-903. 10.1016/j.na.2009.07.021MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ: Variational formulation of a damped Dirichlet impulsive problem. Appl. Math. Lett. 2010, 23: 940-942. 10.1016/j.aml.2010.04.015MathSciNetView ArticleMATHGoogle Scholar
- Xiao J, Nieto JJ: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst. 2011, 348: 369-377. 10.1016/j.jfranklin.2010.12.003MathSciNetView ArticleMATHGoogle Scholar
- Xiao J, Nieto JJ, Luo Z: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 426-432. 10.1016/j.cnsns.2011.05.015MathSciNetView ArticleMATHGoogle Scholar
- Tian Y, Ge WG: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 2008, 51: 509-527.MathSciNetView ArticleMATHGoogle Scholar
- Tian Y, Ge WG: Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 2010, 72: 277-287. 10.1016/j.na.2009.06.051MathSciNetView ArticleMATHGoogle Scholar
- Zhou J, Li Y: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 2010, 72: 1594-1603. 10.1016/j.na.2009.08.041MathSciNetView ArticleMATHGoogle Scholar
- Sun J, Chen H, Nieto JJ, Otero-Novoa M: Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 2010, 72: 4575-4586. 10.1016/j.na.2010.02.034MathSciNetView ArticleMATHGoogle Scholar
- Sun J, Chen H, Nieto JJ: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 2011, 54: 544-555. 10.1016/j.mcm.2011.02.044MathSciNetView ArticleMATHGoogle Scholar
- Wu X, Chen S, Teng K: On variational methods for a class of damped vibration problems. Nonlinear Anal. 2008, 68: 1432-1441. 10.1016/j.na.2006.12.043MathSciNetView ArticleMATHGoogle Scholar
- Duan S, Wu X: The local linking theorem with an application to a class of second-order systems. Nonlinear Anal. 2010, 72: 2488-2498. 10.1016/j.na.2009.10.045MathSciNetView ArticleMATHGoogle Scholar
- Han ZQ, Wang SQ: Multiple solutions for nonlinear systems with gyroscopic terms. Nonlinear Anal. 2012, 75: 5756-5764. 10.1016/j.na.2012.05.020MathSciNetView ArticleMATHGoogle Scholar
- Han Z, Wang S, Yang M: Periodic solutions to second order nonautonomous differential systems with gyroscopic forces. Appl. Math. Lett. 2011, 24: 1343-1346. 10.1016/j.aml.2011.03.005MathSciNetView ArticleMATHGoogle Scholar
- Li X, Wu X, Wu K: On a class of damped vibration problems with super-quadratic potentials. Nonlinear Anal. 2010, 72: 135-142. 10.1016/j.na.2009.06.044MathSciNetView ArticleMATHGoogle Scholar
- Zhang X: Infinitely many solutions for a class of second-order damped vibration systems. Electron. J. Qual. Theory Differ. Equ. 2013., 2013: Article ID 15Google Scholar
- Rabinowitz PH CBMS Regional Conf. Ser. in Math. 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.View ArticleGoogle Scholar
- Bartolo P, Benci V, Fortunato D: Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 1983, 7: 241-273.MathSciNetView ArticleMATHGoogle Scholar
- Luo Z, Xiao J, Xu Y: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations. Nonlinear Anal. 2012, 75: 2249-2255. 10.1016/j.na.2011.10.023MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.