Dependence of Eigenvalues of a Class of Higher-Order Sturm-Liouville Problems on the Boundary

Yang, Qiuxia; Wang, Wanyi; Gao, Xingchao

doi:https://doi.org/10.1155/2015/686102

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2015 | Article ID 686102 | https://doi.org/10.1155/2015/686102

Dependence of Eigenvalues of a Class of Higher-Order Sturm-Liouville Problems on the Boundary

Qiuxia Yang,¹Wanyi Wang,²and Xingchao Gao³

Academic Editor: Sellakkutti Rajendran

Received11 Mar 2014

Accepted09 Sept 2014

Published03 Mar 2015

Abstract

We show that the eigenvalues of a class of higher-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the th eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all th-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest) eigenvalue.

1. Introduction

Dauge and Helffer in [1, 2] considered the second-order Sturm-Liouville (SL) problems and obtained the equations for the eigenvalues of self-adjoint separated boundary conditions. In addition, they showed that the lowest Dirichlet eigenvalue is a decreasing function of the endpoints and thus must have a finite or infinite limit as the end-points approach each other but left open the question of whether this limit is finite or infinite. In [3] the authors showed that it is infinite.

Following the above, Ge et al. in [4] considered the fourth-order Sturm-Liouville differential equation with , and . They showed that its Neumann eigenvalues and Dirichlet eigenvalues, as functions of an endpoint, satisfy the same differential equation form as [1, 2] and the equation for the eigenvalues of self-adjoint separated boundary conditions In particular, they also proved that the lowest Dirichlet eigenvalue is a decreasing function of the endpoints and thus have infinite limit as the endpoints approach each other.

In this paper, partly motivated by the work of Ge et al. in [4], we continue to consider the dependence of eigenvalues of more general form and higher th-order Sturm-Liouville problems on the boundary and also show that the eigenvalues depend not only continuously but also smoothly on boundary points and that the th-order Dirichlet eigenvalues, as functions of the endpoint , satisfy a differential equation of the form We also find the equation satisfied by the th-order Neumann eigenvalues and the equation for the eigenvalues of self-adjoint separated boundary conditions, Furthermore, we prove that as the length of the interval shrinks to zero all higher th-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest) eigenvalue. Although we use the same method of proof as in [4] to get our main results, the specific process of calculation and proof is not completely the same as in [4]. Besides that our conclusions are more concrete and general, theoretical importance, the dependence of the eigenvalues on the interval is fundamental from the numerical point of view (see, e.g., [1–9]).

In Section 2, we summarize some of the basic results needed later and establish the notation. The main results of fourth-order Sturm-Liouville problem are given in Section 3. In Section 4, we consider higher th-order Sturm-Liouville problems and obtain more important results. The last section involves some interesting description about Sturm-Liouville-type boundary value problems.

2. Notation and Basic Results

Consider the differential equation where

We introduce the quasi derivatives of a function , , as follows: then in (6) may be simply written by In this way, the differential expression on is defined for all functions such that exist and are absolutely continuous over compact subintervals of .

Let and consider boundary conditions (BC) where the complex matrices and satisfy

A SL boundary value problem consists of (6) together with boundary conditions (BC) (11). With conditions (7), (10), and (12) it is well known that problem (6), (11) is a regular th-order self-adjoint SL problem which has an infinite but countable number of only real eigenvalues.

From [10], these self-adjoint boundary conditions (11)-(12) are divided into three disjoint subclasses: separated, coupled, and mixed. In the separated case, there are many forms for the th-order problems. In this paper, we only study one form of them.

Consider the following boundary conditions (BC):Here we fix , and the boundary condition (constants), and one endpoint and study the dependence of the eigenvalues and eigenfunctions on the other endpoint.

By a solution of (6) on we mean a function and (6) is satisfied a.e. on . Here denotes the set of functions which are absolutely continuous on all compact subintervals of .

It is well known that the th-order SL boundary value problem consisting of (6) together with boundary conditions (BC) (13a)–(13c), (14a)–(14c) is a regular th-order self-adjoint boundary value problem which has an infinite but countable number of only real eigenvalues. If , a.e. on , then the eigenvalues are bounded below and can be ordered to satisfy

Notation. Let ; for the fourth-order or higher-order Dirichlet and Neumann eigenvalues we use the special notation

By a normalized eigenfunction of the BVP (6), (13a)–(14c), we mean an eigenfunction that satisfies For fixed and fixed boundary condition constants , we abbreviate the notation to and study as a function of for fixed , as varies in the interval .

In the following, we present a continuity result for the eigenvalues and eigenfunctions.

Lemma 1. Let self-adjoint boundary value problems be described as (6), (13a)–(14c). Fix the BC and the endpoint or . Fix . Let for . Then(1) is a continuous function of for .(2)If is simple for some then is simple for every .(3)There exists a normalized eigenfunction of for such that, are uniformly convergent in on any compact subinterval of ; that is, and this convergence is uniform on any compact subinterval of .

Proof. See the proof of Theorem 3 in [3].

Lemma 2. Assume and are solutions of (6) with and , respectively. Then

Proof. This follows from integration by parts.

Lemma 3. Assume a real valued function . Then

Proof. See the proof given in [3].

3. Eigenvalues of Fourth-Order Sturm-Liouville Problem

In this section, we obtain the differentiability of the eigenvalues of the fourth-order boundary value problem, establish differential equations satisfied by them, and discuss the behavior of the Dirichlet eigenvalues as functions of the endpoint .

Consider the differential equation where Let and consider the following boundary conditions (BC) where are quasiderivative. Fix and the boundary condition (constants) and one endpoint and study the dependence of the eigenvalues and eigenfunctions on the other endpoint.

Theorem 4 (fourth-order Dirichlet eigenvalue-eigenfunction differential equation). Let (22) hold. Consider the BVP (21), (23a)–(24b), with and , that is, arbitrary separated conditions at and the fourth-order Dirichlet conditions at . Using the notation of Section 2 and letting , , we have the following differential equation: In particular, if is continuous at and , then (25) holds at .

Proof. For small , in (19), choose , , , and , . From (19) and the boundary conditions (23a)–(24b), noting that , and , we have Since By Lemmas 1 and 3, we have Also from we have And we can obtain Plugging (28), (30), and (31) into (26) divided by and taking the limit as , we get (25). The second part of the theorem follows from above.

Theorem 5 (fourth-order Neumann eigenvalue-eigenfunction differential equation). Let (22) hold. Consider the BVP (21), (23a)–(24b), with and , that is, arbitrary separated conditions at and the fourth-order Neumann conditions at . Using the notation of Section 2 and letting , , we have the following differential equation: In particular, if , and are continuous at , then (32) holds at .

Proof. The proof is similar to Theorem 4. For small , we choose , , and , . From (19) and the boundary conditions (23a)–(24b), noting that , and , we have By Lemmas 1 and 3 we have In a similar way, we have Combining , we also can get When , noting that and plugging (35)–(38) into (33), then we obtain (32). The second part of the theorem follows from the above.

Theorem 6 (eigenvalue-eigenfunction differential equation for separated BVPs). Let (22) hold. Consider the BVP (21), (23a)–(24b), with , , that is, arbitrary separated conditions at and . Using the notation of Section 2 and letting , , we have the following differential equations: Furthermore, if , then If , then In particular, if , , and and are continuous at and , then (39)–(41) hold at .

Proof. The proof is more complicated but consists basically of combining the techniques in the proofs of Theorems 4 and 5. For small , we choose , , and , . From (19) and the boundary conditions (23a)–(24b), noting that , we have Now dividing (42) by and taking the limit as , plugging (28), (30), (35), and (36) into (42), and using the continuity of at , the uniform convergence of to , and Lemma 3, we obtain (39). In addition, from the boundary conditions (24a)–(24b) we note that if then and if then ; plugging them into (39) we obtain (40) and (41).

It is easy to see that Theorem 6 includes Theorems 4 and 5.

Theorem 7. Let (22) hold. Fix and consider the fourth-order Dirichlet eigenvalues for in defined as in (16). If then, for , is strictly decreasing on and

Proof. The decreasing property of as a function of follows directly from Theorem 4. Assume (44) is false, and then by Theorem 4 has a finite limit, say , as and hence is bounded on for . Let be an eigenfunction of normalized to satisfy Next we show that To see this, we first show there exists at least one point such that . Noting that and according to the Rolle’s theorem we know, there exists at least one point such that . Similarly, noting that , hence there exist at least two points and , such that . Therefor there exists at least one point , such that . Using , the boundedness of and the Schwarz inequality, we get So For , we have Thus Noting that as , by (46) and the continuous dependence of solutions (21) on initial conditions and on the parameter we conclude that uniformly on any compact subinterval of . Therefore, for , there exists a , such that This implies that for sufficiently small. This contradicts the normalization (45), which completes the proof.

4. Eigenvalues of Higher-Order Sturm-Liouville Problem

In this section, we obtain the differentiability of the eigenvalues of the th-order boundary value problem and establish differential equations satisfied by them and discuss the behavior of th-order Dirichlet eigenvalueas functions of the endpoint .

Theorem 8 (th-order Dirichlet eigenvalue-eigenfunction differential equation). Let (7) hold. Consider the BVP (6), (13a)–(14c), with and , that is, arbitrary separated conditions at and the th-order Dirichlet conditions at . Using the notation of Section 2 and letting , , we have the following differential equation: In particular, if is continuous at and , then (53) holds at .

Proof. For small , in (19), choose , , and , . From (19) and the boundary conditions (13a)–(14c), noting that , , we have Hence So by Lemmas 1 and 3, we have Similarly, from we can have In addition, noting that Plugging (56), (58), and (59) into (54) divided by and taking the limit as , we get (53). The second part of the theorem follows from above.

Theorem 9 (th-order Neumann eigenvalue-eigenfunction differential equation). Let (7) hold. Consider the BVP (6), (13a)–(14c), with and , that is, arbitrary separated conditions at and the th-order Neumann conditions at . Using the notation of Section 2 and letting , , we have the following differential equation: In particular, if and are continuous at , then (60) holds at .

Proof. The proof is similar to Theorem 8. For small , we choose , , and , . From (19) and the boundary conditions (13a)–(14c), noting that , , , we have By Lemmas 1 and 3 we have In a similar way, we have Combining , we also can get When , noting that and plugging (63)–(66) into (61), then we obtain (60). The second part of the theorem follows from the above.

Theorem 10 (eigenvalue-eigenfunction differential equation for separated BVPs). Let (7) hold. Consider the BVP (6), (13a)–(14c), with , , that is, arbitrary separated conditions at and . Using the notation of Section 2 and letting , , we have the following differential equations: Furthermore, if , then If , then In particular, if and are continuous at and , then (67)–(69) hold at .

Proof. The proof is more complicated but consists basically of combining the techniques in the proofs of Theorems 8 and 9. The concrete process is omitted.

It is easy to see that Theorem 10 includes Theorems 8 and 9.

Theorem 11. Let (7) hold. Fix and consider the th-order Dirichlet eigenvalues for in defined as in (16). If then, for , is strictly decreasing on and

Proof. The decreasing property of as a function of follows directly from Theorem 8. Assume (71) is false, and then by Theorem 8 has a finite limit, say , as , and hence is bounded on for . Let be an eigenfunction of normalized to satisfy First we show that To see this, we first show there exists at least one point such that . Noting that , , and according to the Rolle’s theorem we know, there exists at least one point such that . In the same way, there exist at least points , , such that , , . So there exist at least points , , , , such that . Thus by the Rolle’s theorem we can get the conclusion that there exist at least points , such that . In addition using the boundedness of and the Schwarz inequality, we get So Next we show that For , , also according to the Rolle’s theorem we know, there exist at least zero points such that . Let be the first zero point of . For , we have Thus Noting that as , by (73), (78) and the continuous dependence of solutions (6) on initial conditions and on the parameter we conclude that uniformly on any compact subinterval of . Therefore, for , there exists a , such that This implies that for sufficiently small. This contradicts the normalization (72), which completes the proof.

5. Conclusion

With a simple analysis, we showed that the eigenvalues of a class of th-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the th eigenvalue as a function of an endpoint satisfies a first order differential equation. It is satisfying that these equations are established without any smoothness assumptions on the coefficients and also for the case that the leading coefficient is not assumed to be bounded away from zero and is even allowed to change sign. More importantly, we show that the lowest Dirichlet eigenvalue is a decreasing function of the endpoints and has an infinite limit as the endpoints approach each other.

In recent years, the various physics applications of this kind Sturm-Liouville problem are found in much literature (see, e.g., [11–15]). Many topics in mathematical physics require the investigation of the eigenvalues and eigenfunctions of Sturm-Liouville-type boundary value problems. Our results contain all the cases when is equal to certain special positive integer. In particular, for , Theorem 11 explains that natural frequency of the rod will increase with the shortening of its length.

Furthermore, highly important results in this field have been obtained for the case when the eigenparameter appears not only in the differential equation with transmission conditions but also in the boundary conditions. Particularly, on computing eigenvalues of these types Sturm-Liouville problems, we can refer to [16–18]. Therefore, our proof methods and results will be useful to resolve eigenvalue problem of discontinuous Sturm-Liouville operators and differential operators with eigenparameter boundary conditions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for his/her careful reading of the paper and for making suggestions which have improved the presentation of the paper. The work of the first and third authors is supported by the Talent Introduction Project of Dezhou University (Grant no. 311694) and the work of the second author is supported by the National Nature Science Foundation of China (Grant no. 11361039).

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Copyright © 2015 Qiuxia Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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