- Open Access
Existence of solutions of a second-order impulsive differential equation
© Luo et al.; licensee Springer. 2014
- Received: 10 November 2013
- Accepted: 4 April 2014
- Published: 6 May 2014
This paper is concerned with the existence of solutions of a second-order impulsive differential equation with mixed boundary condition. We obtain sufficient conditions for the existence of a unique solution, at least one solution, at least two solutions and infinitely many solutions, respectively, by using critical point theorems. The main results are also demonstrated with examples.
MSC:34B15, 34B18, 34B37, 58E30.
- impulsive differential equation
- critical point theory
- existence of solutions
where , , , , , α, β are constants with , , and the operator Δ is defined as , where denotes the right-hand (left-hand) limit of at .
In recent years, some classical tools such as some fixed point theorems in cones, topological degree theory and the upper and lower solutions method combined with the monotone iterative technique [9–14] have been widely used to get solutions of impulsive differential equations. On the other hand, in the last few years, some researchers have studied the existence of solutions for impulsive differential equations with boundary conditions via variational methods [15–19]. In this paper, we consider (1.1) by using critical point theory and variational methods.
The rest of this paper is organized as follows. In Section 2 we present several important lemmas. In Section 3, we present existence results of equation (1.1) by using critical point theory and variational methods.
In the following, we first introduce some notations and some necessary definitions.
Definition 2.1 
Let X be a real reflexive Banach space. For any sequence , if is bounded and as possesses a convergent subsequence, then we say that φ satisfies the Palais-Smale condition (PS condition).
Definition 2.2 
Let be differentiable and . We say that φ satisfies the condition if the existence of a sequence in X, such that , as , implies that c is a critical value of φ.
It is clear that the PS condition implies the condition for each .
Lemma 2.3 
Moreover, if a is also symmetric, then the functional defined by attains its minimum at u.
Lemma 2.4 
If φ is weakly lower semi-continuous on a reflexive Banach space X and has a bounded minimizing sequence, then φ has a minimum on X. The existence of a bounded minimizing sequence will be in particular insured when φ is coercive, i.e., such that if .
Lemma 2.5 
X is a real reflexive Banach space;
M is bounded and weakly sequentially closed;
F is weakly sequentially lower semi-continuous on M, i.e., by definition, for each sequence in M such that as , we have .
Lemma 2.6 
If φ satisfies the , then c is a critical value of φ and .
Lemma 2.7 
There exist constants such that ;
For each finite dimensional subspace , there is an such that for every with ;
then φ possesses an unbounded sequence of critical values.
For , we have that u and are both absolutely continuous and , hence for any . If , then u is absolutely continuous and . In this case, the one-side derivatives and may not exist. So, by a classical solution of (1.1), we mean a function satisfying the following conditions: For every , ; u satisfies the boundary condition of (1.1) and the first equation of (1.1); and , , exist and the impulsive conditions of (1.1) hold.
for any . Obviously, is continuous, and a critical point of φ gives a weak solution of (1.1).
Lemma 2.8 If the function is a critical point of the functional φ, then u is a solution of system (1.1).
Thus, u satisfies the equation in (1.1).
If , then .
This contradicts (2.10), so u satisfies the boundary condition. Therefore, u is a solution of system (1.1). □
Lemma 2.9 If , then , where .
Proof The proof follows easily from the Hölder inequality. The detailed argument is similar to the proof of Lemma 2.2 in , and we thus omit it here. □
3.1 Existence of a unique solution
In this section we derive conditions under which system (1.1) admits a unique solution.
Theorem 3.1 Assume that () are fixed constants, and (), then system (1.1) has a unique solution u, and u minimizes the functional (2.2).
It is evident that a is continuous and symmetric and l is bounded. Moreover, a is coercive. By Lemma 2.3, system (1.1) has a unique solution u, and u minimizes the functional (2.2). □
Here , , , , , , . Applying Theorem 3.1, problem (3.1) has a unique solution. By simple calculations, we obtain , , , .
3.2 Existence of at least one solution
In this section we derive conditions under which system (1.1) admits at least one solution. For this purpose, we introduce the following assumption.
Theorem 3.2 Assume that (H1) is satisfied, then system (1.1) has at least one solution u, and u minimizes the functional (2.2).
for all . This implies that φ is coercive.
Let be a weakly convergent sequence to u in H, then converges uniformly to u in .
then . So is weakly sequentially continuous. Clearly, is continuous and convex, which implies that is weakly sequentially lower semi-continuous. Therefore, φ is weakly sequentially lower semi-continuous on H.
By Lemma 2.4, the functional φ has a minimum which is a critical point of φ. Hence, system (1.1) has at least one solution. □
Here , , , , , , . Clearly, (H1) is satisfied. Applying Theorem 3.2, problem (3.2) has at least one solution.
3.3 Existence of at least two distinct solutions
In this section, we derive some sufficient conditions under which the functional φ admits at least two distinct critical points; consequently, (1.1) admits at least two distinct solutions. We first introduce some assumptions.
(H2) uniformly for , .
Theorem 3.3 Assume that (H2) and (H3) are satisfied. Then system (1.1) has at least two solutions.
Proof The proof will be given in three steps.
Step 1. The functional φ satisfies the PS condition.
Hence, is bounded in H.
So (3.5), (3.6) and (3.7) yield in H, i.e., strongly converges to u in H. Therefore, the functional φ satisfies the PS condition.
Step 2. We show that there exists such that the functional φ has a local minimum .
Firstly, we claim that is bounded and weakly sequentially closed.
such that in H. and , since is a closed convex set.
Secondly, we claim that the functional φ is weakly sequentially lower semi-continuous on .
then . By on H, we see that uniformly converges to u in . So is weakly sequentially continuous. Clearly, is continuous and convex, which implies that is weakly sequentially lower semi-continuous. Therefore, φ is weakly sequentially lower semi-continuous on .
Thirdly, we claim that φ has a minimum .
In fact, H is a reflexive Banach space, is bounded and weakly sequentially closed and φ is weakly sequentially lower semi-continuous on . So, by Lemma 2.4, there exists such that .
Finally, we claim that .
Choose , , then for any . Besides, for any . So . Hence, φ has a local minimum .
Step 3. We prove that there exists with such that .
Since , (3.10) implies . Therefore, we can choose with sufficiently large such that .
By Lemma 2.6, c is a critical value of φ, that is, there exists a critical point . Therefore, , are two critical points of φ, and they are solutions of (1.1). □
Here , , , , , , . Let , . Clearly, (H2) and (H3) are satisfied. By Theorem 3.3, problem (3.11) has at least two solutions.
3.4 Existence of infinitely many solutions
In this section, we derive some conditions under which system (1.1) admits infinitely many distinct solutions. To this end, we need the following assumption.
(H4) and , , are odd about u.
Theorem 3.4 Assume that (H2), (H3) and (H4) are satisfied. Then system (1.1) has infinitely many solutions.
Proof We apply Lemma 2.7 to finish the proof. Clearly, is even since and are odd about u, and . The arguments of Theorem 3.3 show that the functional φ satisfies the PS condition. In the same way as in Theorem 3.3, we can easily verify that conditions (i) and (ii) of Lemma 2.7 are satisfied. According to Lemma 2.7, φ possesses infinitely many critical points, i.e., system (1.1) has infinitely many solutions. □
Here , , , , , , . Obviously, , are odd about u. Let , . Clearly, (H2) and (H3) are satisfied. Applying Theorem 3.4, problem (3.12) has infinitely many solutions.
The authors are very grateful to the referees for their very helpful comments and suggestions, which greatly improved the presentation of this paper. This work is supported by the Scientific Research Fund of Hunan Provincial Education Department (No: 13K029).
- Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical, Harlow; 1993.Google Scholar
- Lakshnikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleGoogle Scholar
- Ding W, Mi JR, Han MA: Periodic boundary value problems for the first order impulsive functional differential equations. Appl. Math. Comput. 2005, 165: 433–446. 10.1016/j.amc.2004.06.022MathSciNetView ArticleGoogle Scholar
- Hristova SG, Kulev GK: Quasilinearization of a boundary value problem for impulsive differential equations. J. Comput. Appl. Math. 2001, 132: 399–407. 10.1016/S0377-0427(00)00442-8MathSciNetView ArticleGoogle Scholar
- Luo ZG, Nieto JJ: New results for the periodic boundary value problems for impulsive integro-differential equations. Nonlinear Anal. 2009, 70: 2248–2260. 10.1016/j.na.2008.03.004MathSciNetView ArticleGoogle Scholar
- Li JL: Periodic boundary value problems for second order impulsive integro-differential equations. Appl. Math. Comput. 2008, 198: 317–325. 10.1016/j.amc.2007.08.079MathSciNetView ArticleGoogle Scholar
- Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, London; 1985.Google Scholar
- Rachunkova I, Tvrdy M: Existence results for impulsive second-order periodic problems. Nonlinear Anal. 2005, 59: 133–146.MathSciNetView ArticleGoogle Scholar
- Chen L, Sun J: Nonlinear boundary vale problem of first order impulsive functional differential equations. J. Math. Anal. Appl. 2006, 318: 726–741. 10.1016/j.jmaa.2005.08.012MathSciNetView ArticleGoogle Scholar
- He ZM, He XM: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput. Math. Appl. 2004, 48: 73–84. 10.1016/j.camwa.2004.01.005MathSciNetView ArticleGoogle Scholar
- Nieto JJ, Regan DO: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680–690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleGoogle Scholar
- Qian DB, Li XY: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 2005, 303: 288–303. 10.1016/j.jmaa.2004.08.034MathSciNetView ArticleGoogle Scholar
- Rachunkova I, Tvrdy M: Non-order lower and upper function in second order impulsive periodic problems. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2005, 12: 397–415.MathSciNetGoogle Scholar
- Zuo WJ, Jiang DQ, Regan DO, Agarwal RP: Optimal existence conditions for the periodic delay Φ-Laplace equation with upper and lower solutions in the reverse order. Results Math. 2003, 44: 375–385. 10.1007/BF03322992MathSciNetView ArticleGoogle 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 ArticleGoogle Scholar
- Tian Y, Wang J, Ge WG: Variational methods to mixed boundary value problem for impulsive differential equations. Taiwan. J. Math. 2009, 13: 1353–1370.MathSciNetGoogle Scholar
- Xie JL, Luo ZG: Solutions to a boundary value problem of a fourth-order impulsive differential equation. Bound. Value Probl. 2013., 2013: Article ID 154Google Scholar
- Zhang H, Li ZX: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl. 2010, 11: 67–78. 10.1016/j.nonrwa.2008.10.016MathSciNetView ArticleGoogle Scholar
- Zhang ZH, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155–162. 10.1016/j.nonrwa.2008.10.044MathSciNetView ArticleGoogle Scholar
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.View ArticleGoogle Scholar
- Chipot M: Elements of Nonlinear Analysis. Birkhäuser, Basel; 2000.View ArticleGoogle Scholar
- Kaus D: Nonlinear Functional Analysis. Dover Publications, Dover; 2009.Google Scholar
- Rabinowitz PH CBMS Regional Conference Series in Mathematics 65. In Minimax Methods in Critical Point Theory with Applications to Differential Equations. Am. Math. Soc., Providence; 1986.Google Scholar
- Zhou JW, Li YK: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 2009, 71: 2856–2865. 10.1016/j.na.2009.01.140MathSciNetView ArticleGoogle 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 credited.