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Multiplicity of solutions for second-order impulsive differential equations with Sturm-Liouville boundary conditions
Advances in Difference Equations volume 2014, Article number: 49 (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.
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].
In this paper, we consider the following second-order impulsive differential equations with Sturm-Liouville boundary conditions:
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.
More precisely, Tian and Ge  studied a linear impulsive problem with Sturm-Liouville boundary conditions:
and a nonlinear impulsive problem:
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.
(H1) There exist , and , such that for all ,
(H2) uniformly for , , .
(H3) and are odd with respect to u.
2 Preliminaries and statements
Firstly, we introduce some notations and some necessary definitions.
In the Sobolev space , consider the inner product
inducing the norm
We also consider the inner product
and the norm
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 , .
We multiply the two sides of the first equation of (1.1) by and integrate from 0 to 1, and we have
Considering the above, we need to introduce a different concept of solution for problem (1.1).
Definition 2.1 We say that a function is a weak solution of problem (1.1) if the identity
holds for any .
We consider the functional , defined by
where . Using the continuity of f and , , one has . For any , we have
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).
Proof The proof is similar to . For any and with , for every . Then
By the definition of weak derivative, the above equality implies
Hence and u satisfies the first equation of (1.1) a.e. on .
Now, multiplying by and integrating between 0 and 1, we get
Next we will show that u satisfies the impulsive conditions in (1.1). If not, without loss of generality, we assume that there exists such that
Let , then by (2.5), we get
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). □
Lemma 2.2 Let , then
Proof By using the same methods as , we can obtain the result, here we omit it. □
then we have the following.
Lemma 2.3 If and , or and , there exist constants such that
Proof Firstly we prove the left part of (2.8),
If and , then .
If and , then
If and , then
From (i), (ii), and (iii), set , and we have
On the other hand,
Set , 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 )
Let , and let φ satisfy the P.S. condition. Assume that there exist and a bounded neighborhood Ω of such that is not in Ω and
Then there exists a critical point u of φ, i.e., , with
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 )
Let E be an infinite dimensional real Banach space. Let be an even functional which satisfies the P.S. condition, and . Suppose that , where V is finite dimensional, and φ satisfies:
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 )
For the functional with M not empty, has a solution in case the following hold:
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 .
3 Main results
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 .
To show that φ is weakly lower semi-continuous, let be a weakly convergent sequence to u in X. Then , and converges uniformly to u in , and
We conclude that . Then φ is weakly lower semi-continuous.
Next we show that φ satisfies the P.S. condition. Let be a bounded sequence such that , then there exists a constant such that
By (2.2) and (2.8), we get
From (2.8) and (3.1), we have
Since is bounded, from (3.2) we see that is bounded.
From the reflexivity of X, we may extract a weakly convergent subsequence, which, for simplicity, we call , in X. In the following we will verify that strongly converges to u. We have
By in X, we see that uniformly converges to u in . So
So we obtain , as . That is, strongly converges to u in X, which means φ satisfies the P.S. condition. □
Lemma 3.2 Assume that (H1) holds, then there exist , , such that
Proof From (H1), we get
Integrating the above two inequalities from 1 to u and u to −1, respectively, we have
So there exists a constant such that
From the continuity of , there exists a constant , such that
It follows from (3.5) and (3.6) that
Using the same methods, we know that there exist two constants and such that
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 .
In fact, by (H2), we can choose , then there exist satisfying
For any , , we have
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 .
Let , . From (3.3) and (3.4), we have
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.
For each finite dimensional subspace , for any and , the inequality
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. □
Example 4.1 Let , , , , , we consider the Sturm-Liouville boundary value problem with impulse
Compared with (1.1), , .
The conditions (H1), (H2) are satisfied. Applying Theorem 3.1, problem (4.1) has at least two solutions.
Example 4.2 Let , , , , , consider the Sturm-Liouville boundary value problem with impulse
Compared with (1.1), , .
The conditions (H1), (H2), (H3) are satisfied. Applying Theorem 3.2, problem (4.2) has infinitely many solutions.
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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).
The authors declare that they have no competing interests.
LY and JL carried out the proof of the main part of this article, ZL corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.
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Yan, L., Luo, Z. & Liu, J. Multiplicity of solutions for second-order impulsive differential equations with Sturm-Liouville boundary conditions. Adv Differ Equ 2014, 49 (2014). https://doi.org/10.1186/1687-1847-2014-49
- variational methods
- impulsive differential equations
- boundary value problem