Variational Method to the Impulsive Equation with Neumann Boundary Conditions
© J. Sun and H. Chen. 2009
Received: 28 August 2009
Accepted: 28 September 2009
Published: 11 October 2009
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.
In this paper, we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects
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 . 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 , the method of upper and lower solutions with monotone iterative technique , 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 , the authors studied the following equation with impulsive effects:
where is continuous, , are continuous, and . They essentially proved that IBVP (1.2) has at least two positive solutions via variational method. Recently, in , using variational method and critical point theory, Nieto and O'Regan studied the existence of solutions of the following equation:
where is continuous, and are continuous. They obtained that IBVP (1.3) has at least one solution. Shortly, in , authors extended the results of IBVP (1.3).
In ,Zhou and Li studied the existence of solutions of the following equation:
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 , in which impulse effects are not involved.
In this paper, we will need the following conditions.
(H2) uniformly for , and .
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.
Take . Then . We transform IBVP (1.1) into the following equivalent form:
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]).
Then is a critical value of ; that is, there exists such that and , where
Theorem 2.3 ().
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
which induces the usual norm
We also consider the inner product
and the norm
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
It is clear that is differentiable at any and
for any . Obviously, is continuous.
If is a critical point of the functional , then is a classical solution of IBVP (2.1).
for all . Since are arbitrary, (2.21) shows that and it implies . Therefore, is a classical solution of IBVP (2.1).
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.
Assume that (H1) and (H2) hold. Moreover, and the impulsive functions are odd about , then IBVP (1.1) has infinitely many classical solutions.
Obviously, is an even functional and . We divide our proof into three parts in order to show Theorem 3.1.
By (3.3), (3.4), we obtain as . That is, strongly converges to in , which means the that P. S. condition holds for .
Take , then
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.
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.
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.
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
Then IBVP (1.1) has infinitely many classical solutions.
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.
Therefore, By Theorem 2.3, possesses infinite many critical points, that is, IBVP (1.1) has infinite many classical solutions.
Assume that the second inequalities in (H1), (H5), and (H6) hold, moreover, one assumes the following.
Then IBVP (1.1) has at least two classical solutions.
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.
In view of (H8), take , we have , for any . So .
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).
Obviously, if is a bounded function, in view of Theorem 3.4, we can obtain the same result.
Suppose that (H4) and (H5) hold. Then IBVP (1.1) has at least one solution.
The proof is similar to that in , and we omit it here.
Suppose that and impulsive functions are bounded, then IBVP (1.1) has at least one solution.
4. Some Examples
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.
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.
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.
This project was supported by the National Natural Science Foundation of China (10871206).
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