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Variational Method to the Impulsive Equation with Neumann Boundary Conditions
Boundary Value Problems volume 2009, Article number: 316812 (2009)
Abstract
We study the existence and multiplicity of classical solutions for second-order impulsive Sturm-Liouville equation with Neumann boundary conditions. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions, and infinitely many solutions under some different conditions, respectively. Some examples are also given in this paper to illustrate the main results.
1. Introduction
In this paper, we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ1_HTML.gif)
where with
and
positive functions,
is a continuous function,
are continuous,
,
and
denote the right and the left limits, respectively, of
at
,
is the right limit of
, and
is the left limit of
.
In the recent years, a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problems (IBVPs), by which a number of chemotherapy, population dynamics, optimal control, ecology, industrial robotics, and physics phenomena are described. For the general aspects of impulsive differential equations, we refer the reader to the classical monograph [1]. For some general and recent works on the theory of impulsive differential equations, we refer the reader to [2–9]. Some classical tools or techniques have been used to study such problems in the literature. These classical techniques include the coincidence degree theory of Mawhin [10], the method of upper and lower solutions with monotone iterative technique [11], and some fixed point theorems in cones [12–14].
On the other hand, in the last two years, some researchers have used variational methods to study the existence of solutions for impulsive boundary value problems. Variational method has become a new powerful tool to study impulsive differential equations, we refer the reader to [15–20]. More precisely, in [15], the authors studied the following equation with impulsive effects:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ2_HTML.gif)
where is continuous,
, are continuous, and
. They essentially proved that IBVP (1.2) has at least two positive solutions via variational method. Recently, in [16], using variational method and critical point theory, Nieto and O'Regan studied the existence of solutions of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ3_HTML.gif)
where is continuous, and
are continuous. They obtained that IBVP (1.3) has at least one solution. Shortly, in [17], authors extended the results of IBVP (1.3).
In [19],Zhou and Li studied the existence of solutions of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ4_HTML.gif)
where is continuous, and
, are continuous. They proved that IBVP (1.4) has at least one solution and infinitely many solutions by using variational method and critical point theorem.
Motivated by the above facts, in this paper, our aim is to study the variational structure of IBVP (1.1) in an appropriate space of functions and obtain the existence and multiplicity of solutions for IBVP (1.1) by using variational method. To the best of our knowledge, there is no paper concerned impulsive differential equation with Neumann boundary conditions via variational method. In addition, this paper is a generalization of [21], in which impulse effects are not involved.
In this paper, we will need the following conditions.
(H1)There is constants such that for every
and
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ5_HTML.gif)
where .
(H2) uniformly for
, and
.
(H3)There exist numbers and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ6_HTML.gif)
(H4)There exist numbers and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ7_HTML.gif)
(H5)There exist numbers and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ8_HTML.gif)
(H6)There exist numbers and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ9_HTML.gif)
This paper is organized as follows. In Section 2, we present some preliminaries. In Section 3, we discuss the existence and multiplicity of classical solutions to IBVP (1.1). Some examples are presented in this section to illustrate our main results in the last section.
2. Preliminaries
Take . Then
. We transform IBVP (1.1) into the following equivalent form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ10_HTML.gif)
Obviously, the solutions of IBVP (2.1) are solutions of IBVP (1.1). So it suffices to consider IBVP (2.1).
In this section, the following theorem will be needed in our argument. Suppose that is a Banach space (in particular a Hilbert space) and
. We say that
satisfies the Palais-Smale condition if any sequence
for which
is bounded and
as
possesses a convergent subsequence in
. Let
be the open ball in
with the radius
and centered at
and
denote its boundary.
Theorem 2.1 ([22, Theorem 38.A]).
For the functional with
has a solution for which the following hold:
(i) is a real reflexive Banach space;
(ii) is bounded and weakly sequentially closed;
(iii) is weakly sequentially lower semicontinuous on
; that is, by definition, for each sequence
in
such that
as
, one has
holds.
Theorem 2.2 ([16, Theorem 2.2]).
Let be a real Banach space and let
satisfy the Palais-Smale condition. Assume there exist
and a bounded open neighborhood
of
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ11_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ12_HTML.gif)
Then is a critical value of
; that is, there exists
such that
and
, where
Theorem 2.3 ([23]).
Let be a real Banach space, and let
be even satisfying the Palais-Smale condition and
. If
, where
is finite dimensional, and
satisfies that
(A1)there exist constants such that
,
(A2)for each finite dimensional subspace , there is
such that
for all
with
.
Then possesses an unbounded sequence of critical values.
Let us recall some basic knowledge. Denote by the Sobolev space
, and consider the inner product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ13_HTML.gif)
which induces the usual norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ14_HTML.gif)
We also consider the inner product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ15_HTML.gif)
and the norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ16_HTML.gif)
then the norm is equivalent to the usual norm
in
. Hence,
is reflexive. We define the norm in
as
and
, respectively.
For , we have that
are absolutely continuous, and
, hence
for any
. If
, then
is absolutely continuous and
. In this case, the one-side derivatives
may not exist. As a consequence, we need to introduce a different concept of solution. We say that
is a classical solution of IBVP (2.1) if it satisfies the equation in IBVP (2.1) a.e. on
, the limits
exist and impulsive conditions in IBVP (2.1) hold,
exist and
. Moreover, for every
satisfy
.
For each , consider the functional
defined on
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ17_HTML.gif)
It is clear that is differentiable at any
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ18_HTML.gif)
for any . Obviously,
is continuous.
Lemma 2.4.
If is a critical point of the functional
, then
is a classical solution of IBVP (2.1).
Proof.
Let be a critical point of the functional
. It shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ19_HTML.gif)
holds for any . Choose any
and
such that
if
for
. Equation (2.10) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ20_HTML.gif)
This means, for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ21_HTML.gif)
where . Thus
is a weak solution of the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ22_HTML.gif)
and therefore Let
, then (2.13) becomes the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ23_HTML.gif)
Then the solution of (2.14) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ24_HTML.gif)
where and
are two constants. Then
and
. Therefore,
is a classical solution of (2.13) and
satisfies the equation in IBVP (2.1) a.e. on
. By the previous equation, we can easily get that the limits
and
exist. By integrating (2.10), one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ25_HTML.gif)
and combining with (2.13) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ26_HTML.gif)
Next we will show that satisfies the impulsive conditions in IBVP (2.1). If not, without loss of generality, we assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ27_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ28_HTML.gif)
Obviously, . Substituting them into (2.17), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ29_HTML.gif)
which contradicts (2.18). So satisfies the impulsive conditions in IBVP (2.1). Thus, (2.17) becomes the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ30_HTML.gif)
for all . Since
are arbitrary, (2.21) shows that
and it implies
. Therefore,
is a classical solution of IBVP (2.1).
Lemma 2.5.
Let . Then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ31_HTML.gif)
Proof.
By using the same methods of [15, Lemma 2.6], we easily obtain the above result, and we omit it here.
3. Main Results
In this section, we will show our main results and prove them.
Theorem 3.1.
Assume that (H1) and (H2) hold. Moreover, and the impulsive functions
are odd about
, then IBVP (1.1) has infinitely many classical solutions.
Proof.
Obviously, is an even functional and
. We divide our proof into three parts in order to show Theorem 3.1.
Firstly, We will show that satisfies the Palais-Smale condition. Let
be a bounded sequence such that
. Then there exists constants
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ32_HTML.gif)
By (2.8), (2.9), (3.1), and (H1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ33_HTML.gif)
It follows that is bounded in
. From the reflexivity of
, we may extract a weakly convergent subsequence that, for simplicity, we call
in
. In the following we will verify that
strongly converges to
in
. By (2.9) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ34_HTML.gif)
By in
, we see that
uniformly converges to
in
. So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ35_HTML.gif)
By (3.3), (3.4), we obtain as
. That is,
strongly converges to
in
, which means the that P. S. condition holds for
.
Secondly, we verify the condition (A1) in Theorem 2.3. Let , then
, where
. In view of (H2), take
, there exists an
such that for every
with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ36_HTML.gif)
Hence, for any with
, by (2.8) and (3.5) , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ37_HTML.gif)
Take , then
Finally, we verify condition (A2) in Theorem 2.3. According to (H1), for any and
we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ38_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ39_HTML.gif)
for all and
. This implies that
for all
and
. Similarly, we can prove that there is a constant
such that
for all
and
. Since
is continuous on
, there exists
such that
on
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ40_HTML.gif)
where .
Similarly, there exist constants such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ41_HTML.gif)
For every and
, by (2.8), (3.9), and (3.10), we have that the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ42_HTML.gif)
holds. Take such that
, since
, (3.11) implies that there exists
such that
and
for
. Since
is a finite dimensional subspace, there exists
such that
on
. By Theorem 2.3,
possesses infinite many critical points; that is, IBVP (1.1) has infinite many classical solutions.
Theorem 3.2.
Assume that (H1) and the first equality in (H2) hold. Moreover, is odd about
and the impulsive functions
are odd and nonincreasing. Then IBVP (1.1) has infinitely many classical solutions.
Proof.
We only verify (A1) in Theorem 2.3. Since are odd and nonincreasing continuous functions, then for any
,
. So we have
. Take
, like in (3.6) we can obtain the result.
Theorem 3.3.
Suppose that the first inequalities in (H1), (H3), and (H4) hold. Furthermore, one assumes that and the impulsive functions
are odd about
and we have the following.
(H7)There exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ43_HTML.gif)
Then IBVP (1.1) has infinitely many classical solutions.
Proof.
Obviously, is an even functional and
. Firstly, we will show that
satisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the first inequalities in (H1) and (H4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ44_HTML.gif)
It follows that is bounded in
. In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.
Secondly, as in Theorem 3.1, we can obtain that condition (A2) in Theorem 2.1 is satisfied.
Take the same direct sum decomposition as in Theorem 3.1. For any
, by (2.8), (H3), and (H4), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ45_HTML.gif)
In view of (H7), set , then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ46_HTML.gif)
Therefore, By Theorem 2.3,
possesses infinite many critical points, that is, IBVP (1.1) has infinite many classical solutions.
Theorem 3.4.
Assume that the second inequalities in (H1), (H5), and (H6) hold, moreover, one assumes the following.
(H8) There exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ47_HTML.gif)
Then IBVP (1.1) has at least two classical solutions.
Proof.
We will use Theorems 2.1 and 2.2 to prove the main results. Firstly, we will show that satisfies the Palais-Smale condition. Similarly, as in the proof of Theorem 3.1, by (2.8), (2.9), (3.1), the second inequalities in (H1) and (H5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ48_HTML.gif)
It follows that is bounded in
. In the following, the proof of P. S. condition is the same as that in Theorem 3.1, and we omit it here.
Let , which will be determined later. Set
, then
is a closed ball. From the reflexivity of
, we can easily obtain that
is bounded and weakly sequentially closed. We will show that
is weakly lower semicontinuous on
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ49_HTML.gif)
Then . By
on
we see that
uniformly converges to
in
. So
is weakly continuous. Clearly,
is continuous, which, together with the convexity of
, implies that
is weakly lower semicontinuous. Therefore,
is weakly lower semi-continuous on
. So by Theorem 2.1, without loss of generality, we assume that
. Now we will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ50_HTML.gif)
For any , by (H5) and (H6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ51_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ52_HTML.gif)
In view of (H8), take , we have
, for any
. So
.
Next we will verify that there exists a with
such that
. Let
. Then by (3.10) and (H5), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ53_HTML.gif)
Since , we have
. Therefore, there exists a sufficiently large
with
such that
. Set
, then
. So by Theorem 2.2, there exists
such that
. Therefore,
and
are two critical points of
, and they are classical solutions of IBVP (1.1).
Remark 3.5.
Obviously, if is a bounded function, in view of Theorem 3.4, we can obtain the same result.
Theorem 3.6.
Suppose that (H4) and (H5) hold. Then IBVP (1.1) has at least one solution.
Proof.
The proof is similar to that in [19], and we omit it here.
Corollary 3.7.
Suppose that and impulsive functions
are bounded, then IBVP (1.1) has at least one solution.
4. Some Examples
Example 4.1.
Consider the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ54_HTML.gif)
where
Obviously, are odd on
. Compared to IBVP (1.1),
. By simple calculations, we obtain that
. Let
. Clearly, (H1), (H2) are satisfied. Applying Theorem 3.1, IBVP (4.1) has infinitely many classical solutions.
Example 4.2.
Consider the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ55_HTML.gif)
where .
Obviously, are odd on
. Compared to IBVP (1.1),
. By simple calculations, we obtain that
. Let
. Clearly, the first inequalities in (H1), (H3), and (H4) are satisfied. Take
, then (H7) is also satisfied. Applying Theorem 3.3, IBVP (4.2) has infinitely many classical solutions.
Example 4.3.
Consider the following problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F316812/MediaObjects/13661_2009_Article_839_Equ56_HTML.gif)
where .
Compared to IBVP (1.1), . By simple calculations, we obtain that
. Let
. Clearly, the second inequalities in (H1), (H5), and (H6) are satisfied. Take
, then (H8) is also satisfied. Applying Theorem 3.4, IBVP (4.3) has at least two classical solutions.
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This project was supported by the National Natural Science Foundation of China (10871206).
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Sun, J., Chen, H. Variational Method to the Impulsive Equation with Neumann Boundary Conditions. Bound Value Probl 2009, 316812 (2009). https://doi.org/10.1155/2009/316812
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DOI: https://doi.org/10.1155/2009/316812