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

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 n 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 2k th-order Dirichlet eigenvalues march off to plus infinity; this is also true for the first (i.e., lowest) eigenvalue.

Qiuxia Yang 1 and Wanyi Wang 2 and Xingchao Gao 3

Academic Editor:Sellakkutti Rajendran

1, Information Management College, Dezhou University, Dezhou 253023, China
2, Mathematics Science College, Inner Mongolia University, Huhhot 010021, China
3, Automobile Engineering College, Dezhou University, Dezhou 253023, China

Received 11 March 2014; Accepted 9 September 2014; 3 March 2015

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.

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 [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] th-order Dirichlet eigenvalues, as functions of the endpoint [figure omitted; refer to PDF] , satisfy a differential equation of the form [figure omitted; refer to PDF] We also find the equation satisfied by the [figure omitted; refer to PDF] th-order Neumann eigenvalues [figure omitted; refer to PDF] and the equation for the eigenvalues of self-adjoint separated boundary conditions, [figure omitted; refer to PDF] Furthermore, we prove that as the length of the interval shrinks to zero all higher [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

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

Let [figure omitted; refer to PDF] and consider boundary conditions (BC) [figure omitted; refer to PDF] where the complex [figure omitted; refer to PDF] matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] th-order problems. In this paper, we only study one form of them.

Consider the following boundary conditions (BC): [figure omitted; refer to PDF] [figure omitted; refer to PDF] Here we fix [figure omitted; refer to PDF] , [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] we mean a function [figure omitted; refer to PDF] and (6) is satisfied a.e. on [figure omitted; refer to PDF] . Here [figure omitted; refer to PDF] denotes the set of functions which are absolutely continuous on all compact subintervals of [figure omitted; refer to PDF] .

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

Notation . Let [figure omitted; refer to PDF] ; for the fourth-order or higher-order Dirichlet and Neumann eigenvalues we use the special notation [figure omitted; refer to PDF]

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

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 [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . Fix [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Then

(1) [figure omitted; refer to PDF] is a continuous function of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

(2) If [figure omitted; refer to PDF] is simple for some [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is simple for every [figure omitted; refer to PDF] .

(3) There exists a normalized eigenfunction [figure omitted; refer to PDF] of [figure omitted; refer to PDF] for [figure omitted; refer to PDF] such that, [figure omitted; refer to PDF] are uniformly convergent in [figure omitted; refer to PDF] on any compact subinterval of [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF]

and this convergence is uniform on any compact subinterval of [figure omitted; refer to PDF] .

Proof.

See the proof of Theorem 3 in [3].

Lemma 2.

Assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are solutions of (6) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. Then [figure omitted; refer to PDF]

Proof.

This follows from integration by parts.

Lemma 3.

Assume a real valued function [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] .

Consider the differential equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] and consider the following boundary conditions (BC) [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are quasiderivative. Fix [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and the fourth-order Dirichlet conditions at [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equation: [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then (25) holds at [figure omitted; refer to PDF] .

Proof.

For small [figure omitted; refer to PDF] , in (19), choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From (19) and the boundary conditions (23a)-(24b), noting that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] By Lemmas 1 and 3, we have [figure omitted; refer to PDF] Also from [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] And we can obtain [figure omitted; refer to PDF] Plugging (28), (30), and (31) into (26) divided by [figure omitted; refer to PDF] and taking the limit as [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and the fourth-order Neumann conditions at [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equation: [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous at [figure omitted; refer to PDF] , then (32) holds at [figure omitted; refer to PDF] .

Proof.

The proof is similar to Theorem 4. For small [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From (19) and the boundary conditions (23a)-(24b), noting that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By Lemmas 1 and 3 we have [figure omitted; refer to PDF] In a similar way, we have [figure omitted; refer to PDF] Combining [figure omitted; refer to PDF] , we also can get [figure omitted; refer to PDF] When [figure omitted; refer to PDF] , noting that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equations: [figure omitted; refer to PDF] Furthermore, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then (39)-(41) hold at [figure omitted; refer to PDF] .

Proof.

The proof is more complicated but consists basically of combining the techniques in the proofs of Theorems 4 and 5. For small [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From (19) and the boundary conditions (23a)-(24b), noting that [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Now dividing (42) by [figure omitted; refer to PDF] and taking the limit as [figure omitted; refer to PDF] , plugging (28), (30), (35), and (36) into (42), and using the continuity of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , the uniform convergence of [figure omitted; refer to PDF] to [figure omitted; refer to PDF] , and Lemma 3, we obtain (39). In addition, from the boundary conditions (24a)-(24b) we note that if [figure omitted; refer to PDF] then [figure omitted; refer to PDF] and if [figure omitted; refer to PDF] then [figure omitted; refer to PDF] ; 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 [figure omitted; refer to PDF] and consider the fourth-order Dirichlet eigenvalues [figure omitted; refer to PDF] for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] defined as in (16). If [figure omitted; refer to PDF] then, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is strictly decreasing on [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Proof.

The decreasing property of [figure omitted; refer to PDF] as a function of [figure omitted; refer to PDF] follows directly from Theorem 4. Assume (44) is false, and then by Theorem 4 [figure omitted; refer to PDF] has a finite limit, say [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] and hence is bounded on [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be an eigenfunction of [figure omitted; refer to PDF] normalized to satisfy [figure omitted; refer to PDF] Next we show that [figure omitted; refer to PDF] To see this, we first show there exists at least one point [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Noting that [figure omitted; refer to PDF] and according to the Rolle's theorem we know, there exists at least one point [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Similarly, noting that [figure omitted; refer to PDF] , hence there exist at least two points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Therefor there exists at least one point [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Using [figure omitted; refer to PDF] , the boundedness of [figure omitted; refer to PDF] and the Schwarz inequality, we get [figure omitted; refer to PDF] So [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , by (46) and the continuous dependence of solutions (21) on initial conditions and on the parameter we conclude that [figure omitted; refer to PDF] uniformly on any compact subinterval of [figure omitted; refer to PDF] . Therefore, for [figure omitted; refer to PDF] , there exists a [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] th-order boundary value problem and establish differential equations satisfied by them and discuss the behavior of [figure omitted; refer to PDF] th-order Dirichlet eigenvalueas functions of the endpoint [figure omitted; refer to PDF] .

Theorem 8 ( [figure omitted; refer to PDF] th-order Dirichlet eigenvalue-eigenfunction differential equation).

Let (7) hold. Consider the BVP (6), (13a)-(14c), with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] th-order Dirichlet conditions at [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equation: [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then (53) holds at [figure omitted; refer to PDF] .

Proof.

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

Theorem 9 ( [figure omitted; refer to PDF] th-order Neumann eigenvalue-eigenfunction differential equation).

Let (7) hold. Consider the BVP (6), (13a)-(14c), with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] th-order Neumann conditions at [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equation: [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous at [figure omitted; refer to PDF] , then (60) holds at [figure omitted; refer to PDF] .

Proof.

The proof is similar to Theorem 8. For small [figure omitted; refer to PDF] , we choose [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . From (19) and the boundary conditions (13a)-(14c), noting that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] By Lemmas 1 and 3 we have [figure omitted; refer to PDF] In a similar way, we have [figure omitted; refer to PDF] Combining [figure omitted; refer to PDF] , we also can get [figure omitted; refer to PDF] When [figure omitted; refer to PDF] , noting that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , that is, arbitrary separated conditions at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Using the notation of Section 2 and letting [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have the following differential equations: [figure omitted; refer to PDF] Furthermore, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] In particular, if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then (67)-(69) hold at [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] and consider the [figure omitted; refer to PDF] th-order Dirichlet eigenvalues [figure omitted; refer to PDF] for [figure omitted; refer to PDF] in [figure omitted; refer to PDF] defined as in (16). If [figure omitted; refer to PDF] then, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is strictly decreasing on [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Proof.

The decreasing property of [figure omitted; refer to PDF] as a function of [figure omitted; refer to PDF] follows directly from Theorem 8. Assume (71) is false, and then by Theorem 8 [figure omitted; refer to PDF] has a finite limit, say [figure omitted; refer to PDF] , as [figure omitted; refer to PDF] , and hence is bounded on [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be an eigenfunction of [figure omitted; refer to PDF] normalized to satisfy [figure omitted; refer to PDF] First we show that [figure omitted; refer to PDF] To see this, we first show there exists at least one point [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Noting that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and according to the Rolle's theorem we know, there exists at least one point [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . In the same way, there exist at least [figure omitted; refer to PDF] points [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . So there exist at least [figure omitted; refer to PDF] points [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . Thus by the Rolle's theorem we can get the conclusion that there exist at least [figure omitted; refer to PDF] points [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] . In addition using the boundedness of [figure omitted; refer to PDF] and the Schwarz inequality, we get [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Next we show that [figure omitted; refer to PDF] For [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , also according to the Rolle's theorem we know, there exist at least [figure omitted; refer to PDF] zero points such that [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the first zero point of [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , by (73), (78) and the continuous dependence of solutions (6) on initial conditions and on the parameter we conclude that [figure omitted; refer to PDF] uniformly on any compact subinterval of [figure omitted; refer to PDF] . Therefore, for [figure omitted; refer to PDF] , there exists a [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] th-order Sturm-Liouville problems depend not only continuously but also smoothly on boundary points and that the derivative of the [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] is equal to certain special positive integer. In particular, for [figure omitted; refer to PDF] , 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.

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).

Conflict of Interests

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

References

[1] M. Dauge, B. Helffer, "Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators," Journal of Differential Equations , vol. 104, no. 2, pp. 243-262, 1993.

[2] M. Dauge, B. Helffer, "Eigenvalues variation, II. Neumann problem for Sturm-Liouville operators," Journal of Differential Equations , vol. 104, pp. 263-297, 1993.

[3] Q. Kong, A. Zettl, "Dependence of eigenvalues of Sturm-Liouville problems on the boundary," Journal of Differential Equations , vol. 126, no. 2, pp. 389-407, 1996.

[4] S. Q. Ge, W. Y. Wang, J. Q. Suo, "Dependence of eigenvalues of a class of fourth-order Sturm-Liouville problems on the boundary," Applied Mathematics and Computation , vol. 220, pp. 268-276, 2013.

[5] Q. Kong, A. Zettl, "Eigenvalues of regular Sturm-Liouville problems," Journal of Differential Equations , vol. 131, no. 1, pp. 1-19, 1996.

[6] Q. Kong, H. Wu, A. Zettl, "Dependence of the n th Sturm-Liouville eigenvalue on the problem," Journal of Differential Equations , vol. 156, no. 2, pp. 328-354, 1999.

[7] Q. Kong, H. Wu, A. Zettl, "Geometric aspects of Sturm-Liouville problems. I. structures on spaces of boundary conditions," Proceedings of the Royal Society of Edinburgh Section A: Mathematics , vol. 130, no. 3, pp. 561-589, 2000.

[8] X. Cao, Q. Kong, H. Wu, A. Zettl, "Sturm-Liouville problems whose leading coefficient function changes sign," Canadian Journal of Mathematics , vol. 55, no. 4, pp. 724-749, 2003.

[9] Q. Kong, H. Wu, A. Zettl, "Limits of Sturm-Liouville eigenvalues when the interval shrinks to an end point," Proceedings of the Royal Society of Edinburgh Section A: Mathematics , vol. 138, no. 2, pp. 323-338, 2008.

[10] A. P. Wang, J. Sun, A. Zettl, "The classification of self-adjoint boundary conditions: separated, coupled, and mixed," Journal of Functional Analysis , vol. 255, no. 6, pp. 1554-1573, 2008.

[11] M. Demirci, Z. Akdogan, O. Sh. Mukhtarov, "Asymptotic behavior of eigenvalues and eigenfuctions of one discontinuous boundary value problem," International Journal of Computational Cogonition , vol. 2, no. 3, pp. 101-113, 2004.

[12] D. Buschmann, G. Stolz, J. Weidmann, "One-dimensional Schrödinger operators with local point interactions," Journal für die Reine und Angewandte Mathematik , vol. 467, pp. 169-186, 1995.

[13] E. H. Doha, A. H. Bhrawy, R. M. Hafez, "A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations," Mathematical and Computer Modelling , vol. 53, no. 9-10, pp. 1820-1832, 2011.

[14] E. H. Doha, A. H. Bhrawy, R. M. Hafez, "A Jacobi dual-Petrov-Galerkin method for solving some odd-order ordinary differential equations," Abstract and Applied Analysis , vol. 2011, 2011.

[15] E. Tohidi, A. H. Bhrawy, K. Erfani, "A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation," Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems , vol. 37, no. 6, pp. 4283-4294, 2013.

[16] M. M. Tharwat, A. H. Bhrawy, A. S. Alofi, "Computing eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions using Hermite approximation," Abstract and Applied Analysis , vol. 2013, 2013.

[17] M. M. Tharwat, A. H. Bhrawy, A. Yildirim, "Numerical computation of eigenvalues of discontinuous Sturm-Liouville problems with parameter dependent boundary conditions using sinc method," Numerical Algorithms , vol. 63, no. 1, pp. 27-48, 2013.

[18] M. M. Tharwat, A. H. Bhrawy, A. S. Alofi, "Approximation of eigenvalues of discontinuous Sturm-Liouville problems with eigenparameter in all boundary conditions," Boundary Value Problems , vol. 2013, article 132, 2013.