- Open Access
Multiplicity of solutions for second-order impulsive differential equations with Sturm-Liouville boundary conditions
© Yan et al.; licensee Springer. 2014
- Received: 8 July 2013
- Accepted: 13 January 2014
- Published: 31 January 2014
In this paper, we use variational methods to investigate the solutions of impulsive differential equations with Sturm-Liouville boundary conditions. The conditions for the existence and multiplicity of solutions are established. The main results are also demonstrated with examples.
- variational methods
- impulsive differential equations
- boundary value problem
Impulsive differential equations arising from the real world describe the dynamics of a process in which sudden, discontinuous jumps occur. Such processes are naturally seen in biology, medicine, mechanics, engineering, chaos theory, and so on. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations [1–8].
where , , , , , for , .
In recent years, boundary value problems for impulsive and Sturm-Liouville equations have been studied extensively in the literature. There have been many approaches to the study of positive solutions of differential equations, such as fixed point theory, topological degree theory and the comparison method [9–14]. On the other hand, many researchers have used variational methods to study the existence of solutions for boundary value problems [15–21]. However, to our knowledge, the study of solutions for impulsive differential equations as (1.1) using variational methods has received considerably less of attention.
They obtained the existence of positive solutions for problems (1.2) and (1.3) by using the variational method.
Inspired by the work , in this paper we use critical point theory and variational methods to investigate the multiple solutions of (1.1). Our main results extend the study made in , in the sense that we deal with a class of problems that is not considered in those papers.
We need the following conditions.
(H2) uniformly for , , .
(H3) and are odd with respect to u.
Firstly, we introduce some notations and some necessary definitions.
Then the norm is equivalent to the usual norm in . Hence, X is reflexive. Denote .
For , we find that u and are both absolutely continuous, and , hence for any . If , then u is absolutely continuous and . In this case, is not necessarily valid for every and the derivative may present some discontinuities. This leads to the impulsive effects. As a consequence, we need to introduce a different concept of solution. We say that is a classical solution of IBVP (1.1) if it satisfies the following conditions: u satisfies the first equation of (1.1) a.e. on ; the limits , , exist and the impulsive condition of (1.1) holds; u satisfies the boundary condition of (1.1); for every , .
Considering the above, we need to introduce a different concept of solution for problem (1.1).
holds for any .
Thus, the solutions of problem (1.1) are the corresponding critical points of φ.
Lemma 2.1 If is a weak solution of (1.1), then u is a classical solution of (1.1).
Hence and u satisfies the first equation of (1.1) a.e. on .
which contradicts (2.6), so u satisfies the impulsive conditions of (1.1). Similarly, u satisfies the boundary conditions. Therefore, u is a classical solution of problem (1.1). □
Proof By using the same methods as , we can obtain the result, here we omit it. □
then we have the following.
If and , then .
- (ii)If and , then
- (iii)If and , then
This is the end of the proof. □
We state some basic notions and celebrated results from critical points theory.
Definition 2.2 Let X be a real Banach space (in particular a Hilbert space) and . φ is said to be satisfying the P.S. condition on X if any sequence for which is bounded and as , possesses a convergent subsequence in X.
Lemma 2.4 (see )
Note that if either or is a critical point of φ, then we obtain the existence of at least two critical points for φ.
Lemma 2.5 (see )
there exist and such that ;
for any finite dimensional subspace , there is an such that for every with .
Then φ possesses an unbounded sequence of critical values.
Lemma 2.6 (see )
X is a real reflexive Banach space;
M is bounded and weak sequentially closed;
F is weak sequentially lower semi-continuous on M, i.e., by definition, for each sequence in M such that as , we have .
To prove our main results, we need the following lemmas.
Lemma 3.1 The function defined by (2.1) is continuous, continuously differentiable and weakly lower semi-continuous. Moreover, if , , or , , and (H1) holds, then φ satisfies the P.S. condition.
Proof From the continuity of f and , , we obtain the continuity and differentiability of φ and .
We conclude that . Then φ is weakly lower semi-continuous.
Since is bounded, from (3.2) we see that is bounded.
So we obtain , as . That is, strongly converges to u in X, which means φ satisfies the P.S. condition. □
This is the end of the proof. □
Now we get the main results of this paper.
Theorem 3.1 Suppose , , or , , and (H1) and (H2) hold, then (1.1) has at least two solutions.
Proof In our case it is clear that , Lemma 3.1 has shown that φ satisfies the P.S. condition.
Firstly, we will show that there exists such that the functional φ has a local minimum .
Let , which will be determined later. Since is a Hilbert space, it is easy to deduce that is bounded and weak sequentially closed. Lemma 3.1 has shown that φ is weak sequentially lower semi-continuous on . So by Lemma 2.6, we know that φ has a local minimum .
Without loss of generality, we assume that . Now we will show that .
So for any . Besides, . Then for any . So . Hence, φ has a local minimum .
Next we will verify that there exists a with such that .
Since , , , then we get . Hence, there exists a sufficiently large with such that . Set , then . Hence, by Lemma 2.4, there exists such that . Therefore, and are two critical points of φ, and they are classical solutions of (1.1). □
Theorem 3.2 Suppose , , or , , and (H1), (H2), and (H3) hold, then (1.1) has infinitely many classical solutions.
Proof By (H3), we know that and are odd about u, then φ is even. Moreover, by Lemma 3.1, we know that , , and φ satisfies the P.S. condition.
Next, we will verify the conditions (i) and (ii) of Lemma 2.5.
Let is a finite dimensional subspace, for any , by (3.7), we can easily verify (i) in the same way as in Theorem 3.1.
holds. Take such that , since , , , (3.8) implies that there exists such that and for every . Since is a finite dimensional subspace, we can choose an such that , .
According to Lemma 2.5, φ possesses infinitely many critical points, i.e., the impulsive problem (1.1) has infinitely many solutions. □
Compared with (1.1), , .
The conditions (H1), (H2) are satisfied. Applying Theorem 3.1, problem (4.1) has at least two solutions.
Compared with (1.1), , .
The conditions (H1), (H2), (H3) are satisfied. Applying Theorem 3.2, problem (4.2) has infinitely many solutions.
The authors are grateful to the referees for their useful suggestions. This work is partially supported by the National Natural Science Foundation of China (No. 71201013), the Provincial Natural Science Foundation of Hunan (No. 11JJ3012).
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleGoogle Scholar
- Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.Google Scholar
- Agarwal RP, Franco D, O’Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequ. Math. 2005, 69: 83-96. 10.1007/s00010-004-2735-9MathSciNetView ArticleGoogle Scholar
- Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 2007, 325: 226-299. 10.1016/j.jmaa.2005.04.005MathSciNetView ArticleGoogle Scholar
- Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 2008, 40: 143-150. 10.1112/blms/bdm110MathSciNetView ArticleGoogle Scholar
- Luo ZG, Li JL, Shen JH: Stability of bidirectional associative memory networks with impulses. Appl. Math. Comput. 2011, 218: 1658-1667. 10.1016/j.amc.2011.06.045MathSciNetView ArticleGoogle Scholar
- Lu W, Ge WG, Zhao Z: Oscillatory criteria for third-order nonlinear difference equation with impulses. J. Comput. Appl. Math. 2010, 234: 3366-3372. 10.1016/j.cam.2010.04.037MathSciNetView ArticleGoogle Scholar
- Hernandez E, Henriquez HR, McKibben MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. TMA 2009, 70: 2736-2751. 10.1016/j.na.2008.03.062View ArticleGoogle Scholar
- Ding W, Han M, Mi J: Periodic boundary value problem for the second order impulsive functional equations. Comput. Math. Appl. 2005, 50: 491-507. 10.1016/j.camwa.2005.03.010MathSciNetView ArticleGoogle Scholar
- Chen J, Tisdell CC, Yuan R: On the solvability of periodic boundary value problems with impulse. J. Math. Anal. Appl. 2007, 331: 902-912. 10.1016/j.jmaa.2006.09.021MathSciNetView ArticleGoogle Scholar
- Boucherif A, Al-Qahtani AS, Chanane B: Existence of solutions for second-order impulsive boundary-value problems. Electron. J. Differ. Equ. 2012., 2012: Article ID 24Google Scholar
- Shen JH, Wang WB: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2008, 69: 4055-4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleGoogle Scholar
- Zhao YL, Chen HB: Multiplicity of solutions to two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 2008, 206: 925-931. 10.1016/j.amc.2008.10.009MathSciNetView ArticleGoogle Scholar
- Zeng G, Chen L, Sun L: Existence of periodic solutions of order on of planar impulsive autonomous system. J. Comput. Appl. Math. 2006, 186: 466-481. 10.1016/j.cam.2005.03.003MathSciNetView ArticleGoogle Scholar
- Nieto JJ, Regan DO: Variational approach to impulsive differential equations. Nonlinear Anal. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022View ArticleGoogle 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 ArticleGoogle Scholar
- Xiao J, Nieto JJ, Luo ZG: 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 ArticleGoogle Scholar
- Yan LZ, Liu J, Luo ZG: Existence of solution for impulsive differential equations with nonlinear derivative dependence via variational methods. Abstr. Appl. Anal. 2013., 2013: Article ID 908062Google Scholar
- Yan LZ, Liu J, Luo ZG: Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Adv. Differ. Equ. 2013., 2013: Article ID 293Google Scholar
- Sun J, Chen H-B, Nieto JJ, Otero-Novoa M: The multiplicity of solutions for perturbed second order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 2010, 72: 4575-4586. 10.1016/j.na.2010.02.034MathSciNetView ArticleGoogle Scholar
- Chen P, Tang XH: Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems. Appl. Math. Comput. 2012, 218: 11775-11789. 10.1016/j.amc.2012.05.027MathSciNetView 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
- Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, Berlin; 1989.View ArticleGoogle 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., Washington; 1986.Google Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. III: Variational Methods and Optimization. Springer, Berlin; 1985.View ArticleGoogle Scholar
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