• Research Article
• Open access
• Published:

Variational Method to the Impulsive Equation with Neumann Boundary Conditions

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

(1.1)

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 [2â€“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 [12â€“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 [15â€“20]. More precisely, in [15], the authors studied the following equation with impulsive effects:

(1.2)

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:

(1.3)

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:

(1.4)

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 ,

(1.5)

where .

(H2) uniformly for , and .

(H3)There exist numbers and such that

(1.6)

(H4)There exist numbers and such that

(1.7)

(H5)There exist numbers and such that

(1.8)

(H6)There exist numbers and such that

(1.9)

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:

(2.1)

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, Theoremâ€‰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, Theoremâ€‰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

(2.2)

Let

(2.3)

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

(2.4)

which induces the usual norm

(2.5)

We also consider the inner product

(2.6)

and the norm

(2.7)

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

(2.8)

It is clear that is differentiable at any and

(2.9)

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

(2.10)

holds for any . Choose any and such that if for . Equation (2.10) implies

(2.11)

This means, for any ,

(2.12)

where . Thus is a weak solution of the following equation:

(2.13)

and therefore Let , then (2.13) becomes the following form:

(2.14)

Then the solution of (2.14) can be written as

(2.15)

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

(2.16)

and combining with (2.13) we get

(2.17)

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

(2.18)

Let

(2.19)

Obviously, . Substituting them into (2.17), we get

(2.20)

which contradicts (2.18). So satisfies the impulsive conditions in IBVP (2.1). Thus, (2.17) becomes the following form:

(2.21)

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

(2.22)

Proof.

By using the same methods of [15, Lemmaâ€‰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

(3.1)

By (2.8), (2.9), (3.1), and (H1), we have

(3.2)

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

(3.3)

By in , we see that uniformly converges to in . So

(3.4)

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 ,

(3.5)

Hence, for any with , by (2.8) and (3.5) , we have

(3.6)

Take , then

Finally, we verify condition (A2) in Theorem 2.3. According to (H1), for any and we have that

(3.7)

Hence

(3.8)

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

(3.9)

where .

Similarly, there exist constants such that

(3.10)

For every and , by (2.8), (3.9), and (3.10), we have that the following inequality:

(3.11)

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

(3.12)

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

(3.13)

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

(3.14)

In view of (H7), set , then we have

(3.15)

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

(3.16)

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

(3.17)

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

(3.18)

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

(3.19)

For any , by (H5) and (H6), we have

(3.20)

Hence

(3.21)

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

(3.22)

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:

(4.1)

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:

(4.2)

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:

(4.3)

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.

References

1. Lakshmikantham V, BaÄ­nov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.

2. Agarwal RP, Franco D, O'Regan D: Singular boundary value problems for first and second order impulsive differential equations. Aequationes Mathematicae 2005, 69(1â€“2):83â€“96. 10.1007/s00010-004-2735-9

3. Ahmad B, Nieto JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(10):3291â€“3298. 10.1016/j.na.2007.09.018

4. Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007, 325(1):226â€“236. 10.1016/j.jmaa.2005.04.005

5. Nieto JJ, RodrÃ­guez-LÃ³pez R: New comparison results for impulsive integro-differential equations and applications. Journal of Mathematical Analysis and Applications 2007, 328(2):1343â€“1368. 10.1016/j.jmaa.2006.06.029

6. SamoÄ­lenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, River Edge, NJ, USA; 1995:x+462.

7. HernÃ¡ndez E, HenrÃ­quez HR, McKibben MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(7):2736â€“2751. 10.1016/j.na.2008.03.062

8. Zhang H, Chen L, Nieto JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Analysis: Real World Applications 2008, 9(4):1714â€“1726. 10.1016/j.nonrwa.2007.05.004

9. Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi, New York, NY, USA; 2006:xiv+366.

10. Qian D, Li X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. Journal of Mathematical Analysis and Applications 2005, 303(1):288â€“303. 10.1016/j.jmaa.2004.08.034

11. Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations. Journal of Mathematical Analysis and Applications 2006, 318(2):726â€“741. 10.1016/j.jmaa.2005.08.012

12. Chen J, Tisdell CC, Yuan R: On the solvability of periodic boundary value problems with impulse. Journal of Mathematical Analysis and Applications 2007, 331(2):902â€“912. 10.1016/j.jmaa.2006.09.021

13. Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bulletin of the London Mathematical Society 2008, 40(1):143â€“150. 10.1112/blms/bdm110

14. Li J, Nieto JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses. Boundary Value Problems 2009, 2009:-12.

15. Tian Y, Ge W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proceedings of the Edinburgh Mathematical Society 2008, 51(2):509â€“527.

16. Nieto JJ, O'Regan D: Variational approach to impulsive differential equations. Nonlinear Analysis: Real World Applications 2009, 10(2):680â€“690. 10.1016/j.nonrwa.2007.10.022

17. Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Analysis: Real World Applications. In press

18. Tian Y, Ge W, Yang D: Existence results for second-order system with impulse effects via variational methods. Journal of Applied Mathematics and Computing 2009, 31(1â€“2):255â€“265. 10.1007/s12190-008-0208-3

19. Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(7â€“8):2856â€“2865. 10.1016/j.na.2009.01.140

20. Zhang H, Li Z: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Analysis: Real World Applications. In press

21. Bonanno G, D'AguÃ¬ G: A Neumann boundary value problem for the Sturm-Liouville equation. Applied Mathematics and Computation 2009, 208(2):318â€“327. 10.1016/j.amc.2008.12.029

22. Zeidler E: Nonlinear Functional Analysis and Its Applications. III. Springer, New York, NY, USA; 1985:xxii+662.

23. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics. Volume 65. American Mathematical Society, Washington, DC, USA; 1986:viii+100.

24. Tian Y, Ge W: Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications. In press

Acknowledgment

This project was supported by the National Natural Science Foundation of China (10871206).

Author information

Authors

Corresponding author

Correspondence to Juntao Sun.

Rights and permissions

Reprints and permissions

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