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.
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.
2 Preliminaries and statements
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 .
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 .
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).
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