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

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


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  0 ,  2 ,  :  = (, ) → R, 1/ 0 ,  2 ∈  loc () and  > 0 a.e. on .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 2th-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 2th-order Dirichlet eigenvalues, as functions of the endpoint , satisfy a differential equation of the form  0   = − ( [] ) 2 . ( We also find the equation satisfied by the 2th-order Neumann eigenvalues and the equation for the eigenvalues of self-adjoint separated boundary conditions, + (  − )  2 − ( [] ) 2  0 . (5) Mathematical Problems in Engineering Furthermore, we prove that as the length of the interval shrinks to zero all higher 2th-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].
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 2th-order Sturm-Liouville problems and obtain more important results.The last section involves some interesting description about Sturm-Liouvilletype boundary value problems.
We introduce the quasi derivatives of a function ,  [] ,  = 0, 1, 2, . . ., 2 as follows: then  in ( 6) may be simply written by In this way, the differential expression  on  is defined for all functions  such that  [0] ,  [1] , . . .,  [2−1] exist and are absolutely continuous over compact subintervals of .
where the complex 2 × 2 matrices  and  satisfy The 2 × 4 matrices ( | ) have full rank, 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 2th-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 2th-order problems.In this paper, we only study one form of them.
Here we fix   ( = 0, 1, 2, . . ., ),  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  [0] ,  [1] , . . .,  [2−1] ∈  loc () and ( 6) is satisfied a.e. on .Here  loc () denotes the set of functions which are absolutely continuous on all compact subintervals of .
It is well known that the 2th-order SL boundary value problem consisting of (6) together with boundary conditions (BC) (13a)-(13c), (14a)-(14c) is a regular 2th-order selfadjoint boundary value problem which has an infinite but countable number of only real eigenvalues.If  0 ≥ 0, a.e. on  = (, ), then the eigenvalues are bounded below and can be ordered to satisfy 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  ∈  0 , as  varies in the interval (, ).
Proof.See the proof of Theorem 3 in [3].
Lemma 2. Assume  and V are solutions of (6) with  =  and  = ], respectively.Then Proof.This follows from integration by parts.

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 .
Proof.The proof is more complicated but consists basically of combining the techniques in the proofs of Theorems 4 and 5.
It is easy to see that Theorem 6 includes Theorems 4 and 5.

Eigenvalues of Higher-Order Sturm-Liouville Problem
In this section, we obtain the differentiability of the eigenvalues of the 2th-order boundary value problem and establish differential equations satisfied by them and discuss the behavior of 2th-order Dirichlet eigenvalueas functions of the endpoint .
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.

Conclusion
With a simple analysis, we showed that the eigenvalues of a class of 2th-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  0 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][12][13][14][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  = 2, 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][17][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.