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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 619358, 11 pages
http://dx.doi.org/10.1155/2013/619358
Research Article

The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received 5 July 2013; Revised 5 September 2013; Accepted 26 September 2013

Academic Editor: Alberto Cabada

Copyright © 2013 Mudan Bai 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.

Abstract

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition for being an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green's function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.

1. Introduction

The Sturm-Liouville (S-L) theory, as an active area of research in pure and applied mathematics, plays an important role in solving many problems in mathematical physics and is concerned in many publications [17]. It is well known that for the classical S-L problems, the solutions or the derivatives of the solutions are continuous on the interval, but these conditions cannot be satisfied in many practical physical problems. So, a class of S-L operators with “discontinuity,” that is, the S-L problems with transmission conditions at an interior point, are concerned by many mathematical and physical researchers [810]. Such conditions are known by various names including transmission conditions [11, 12], interface conditions [1315], jump conditions [16], and discontinuous conditions [17, 18].

In this paper, we consider the following Sturm-Liouville equation: with boundary conditions: and transmission conditions: where is a complex eigenparameter and ; notice that the potential function guarantee and in (4) makes sense (see Theorem 1); all coefficients of the boundary and transmission conditions are real numbers. Throughout this paper, we assume that ,  , and We derive the eigenfunction expansion for Green’s function of the S-L problem (1)–(4) and establish the modified Parseval equality of the S-L problem with very general transmission conditions at one inner point of the finite interval .

The organization of this paper is as follows. After the Introduction, we construct the basic solutions of S-L equation (1) with transmission conditions (4) and obtain the sufficient and necessary condition for being an eigenvalue of the S-L problem in Section 2. In Section 3, the asymptotic formulas for eigenvalues and eigenfunctions of the S-L problem are obtained by using the asymptotic expressions of the solutions, and we prove that the eigenvalues of the S-L problem are bounded below and are countably infinite. Section 4 contains the eigenfunction expansion for Green’s function of the S-L problem (1)–(4) and we establish the modified Parseval equality in the associated Hilbert space by using the eigenfunction expansion for Green’s function.

2. The Basic Solutions and Eigenvalues

We construct the basic solutions of S-L equation (1) with transmission conditions (4) and obtain the sufficient and necessary condition for being an eigenvalue of the S-L problem in this section. At first, we prove the existence of finite limits for all solution of (1) and its derivative at both sides of point in the following theorem.

Theorem 1. Assume the coefficient function in S-L equation (1) is real-valued and Lebesgue integrable on and , that is, ; then for the solution of (1), the limits exist and are finite.

In order to prove Theorem 1, we need the following lemma.

Lemma 2 (see [19]). Let . Let denote the set of matrices with entries from .(1) Suppose that satisfy for some , where . For some and , let be the solution of Then exists and is finite. (2)Suppose that for some . For some and , let be the solution of (8) on . Then, exists and is finite.

Proof of Theorem 1. Let , and let . Then (1) is equivalent to the equation on . From Lemma 2, we know that exists and is finite. This implies that exist and are finite.
Let . Then, (1) is equivalent to the equation on . Since exists and is finite by Lemma 2, the limits exist and are finite.

Let us define a new inner product in as follows, which is associated with transmission conditions (4) and useful to investigate the S-L problem (1)–(4): where . It is easy to verify that is a Hilbert space. For simplicity, it is denoted by . The norm induced by the inner product is denoted by . We consider the S-L problem (1)–(4) in the associated Hilbert space .

Theorem 3 (see [14]). The S-L problem (1)–(4) is self-adjoint.

We construct two basic solutions and of S-L equation (1) by the following procedure. At first, we consider the initial-value problem By virtue of Theorem 1.5 in [20], the problem (13) has a unique solution for each , which is an entire function of for each fixed .

Similarly, the initial-value problem has a solution for each . Moreover, is an entire function of for each fixed . We define a function on by Obviously, satisfies S-L equation (1), the boundary condition (2), and both transmission conditions (4); that is, is a solution of S-L equation (1).

As same as above, the initial-value problem has a solution which is an entire function of for each fixed . Similarly, the initial-value problem also has a unique solution , which is an entire function of for each fixed . Let By (16) and (17), satisfies S-L equation (1) and conditions (3) and (4); that is, is another solution of S-L equation (1).

It is well known that, from the ordinary linear differential equation theory, the Wronskians are independent of the variable .

Lemma 4. The equality holds for each .

Proof. Since the Wronskians are independent of the variable , then by (14) and (17),

Let . Then and is an entire function of .

Theorem 5. The eigenvalues of the S-L problem (1)–(4) coincide with the zeros of the function .

Proof. Let be an eigenvalue of the S-L problem (1)–(4) and let be any corresponding eigenfunction. Let us assume that . Then, . Consequently, each pair of functions and is linearly independent. Therefore, the solution of (1) may be represented in the form where at least one of the constants is not zero. Since satisfies boundary conditions (2) and (3), we obtain ,  . And substituting (20) in condition (4), we get and satisfying the following equations: From (14), the determinant of the matrix of coefficients in the last equations is equal to which is not zero by the assumption. Hence, and . This is a contradiction. Thus, each eigenvalue of the S-L problem (1)–(4) is a zero of the function .
Conversely, let . Then, for all . Consequently, the functions and are linearly dependent. That is, for some . Then So, satisfies condition (2). And the solution satisfies condition (3) and both transmission condition (4) from its construction. Thus would be an eigenfunction of the S-L problem (1)–(4) for the eigenvalue .

Lemma 6. The eigenvalues of the S-L problem (1)–(4) are simple.

Proof. Let . We differentiate the equation with respect to and have . Then By integration by parts and some calculations from (13), (14), (16), and (17) where are the derivatives of with respect to . Since is independent of the variable and the value of and given in (13), So, by (24), (25), and (26), Let be an eigenvalue of the S-L problem (1)–(4). Then is real. Since for the eigenvalue of the S-L problem (1)–(4) from Theorem 5, for some . And from (4), we get . Thus, (27) becomes Hence is simple.

3. Asymptotic Formulas for Eigenvalues and Eigenfunctions

In this section, the asymptotic formulas for eigenvalues and eigenfunctions of the S-L problem (1)–(4) are obtained by using the asymptotic expressions of the solutions. At first, we calculate the asymptotic expressions of the solutions.

Lemma 7. Let , . If , then one has the following asymptotic expressions: as . If , then as .

Proof. The asymptotic formulas for follow from the similar formulas of Lemma 1.7 in [20].

The asymptotic formulas for are as follows.

Lemma 8. Let ,  ; then has the following asymptotic formulas as :(1) if , then (2) if , then (3) if , then (4) if , then for .

Proof. Let ,  .   satisfies (14). By the method of variation of constants, we have satisfying the following integral equation: Let . Substituting (30), for , in (36), we have We consider the case . Let . Multiplying (37) by , we have the following: Let . Then for some . Consequently, as . So, as . Substituting the asymptotic equality in (37) gives (32) for . And the case for is obtained by differentiating (37) and by some similar calculations.
Similarly, we can prove (33), (34), and (35).

When is big enough, the asymptotic formulas for are obtained in Lemma 8. Substituting the asymptotic formulas of and the value of at , which is given in (16), into , we can establish the following lemma.

Lemma 9. Let ,  ; then has the following asymptotic formulas for large enough .(1) If , then (2) If , then (3) If , then (4) If , then (5) If , then (6) If , then (7) If , then (8) If , then

Theorem 10. The eigenvalues of the S-L problem (1)–(4) are bounded below and countably infinite; the unique cluster point is infinity.

Proof. From Theorem 5, the eigenvalues of the S-L problem (1)–(4) are zeros of the function which is an entire function of . And the asymptotic formulas of the function are obtained in Lemma 9. Let where for the case from Lemma 9.
Let ,   and let and let . Then is a closed curve on the plane of . Next, we prove that on . When is big enough, on Similarly, when is big enough, on We obtain So, By Rouche’s Theorem in [21], and have the same zeros interior of as follows:
Letting , we can prove that . Hence, as . Thus, only has finite negative zeros. The proof of other cases in Lemma 9 is similar to the proof of the case .

Theorem 11. Let , be the collection of all eigenvalues of the S-L problem (1)–(4) and let be the corresponding normalized eigenfunctions. Then And is complete in and

Theorem 12. The following asymptotic formulas hold for eigenvalues and eigenfunctions of the S-L problem (1)–(4) for large enough . If , then (2) If , then (3) If , then (4) If , then (5) If , then (6) If , then (7) If , then (8) If , then

Proof. By Theorem 5, we know that the eigenvalues of the S-L problem (1)–(4) coincide with the zeros of the function . From Lemma 9 and Theorem 10, the entire function has a sequence of zeros: for the case when is big enough. By using the asymptotic formulas of eigenvalues , the corresponding eigenfunctions have the following asymptotic formulas: for the case . Similarly, we can obtain the other cases.

By the asymptotic formulas for the eigenvalues and eigenfunctions of the S-L problem (1)–(4) in the above theorem, we know that the series converges absolutely and uniformly where is the normalized eigenfunction.

Green’s function of the classical S-L problem is studied in [22]. Here, we obtain Green’s function of the S-L problem (1)–(4) as follows:

Let us consider the nonhomogeneous differential equation together with boundary and transmission conditions (2)–(4) where . Green’s functions provide a powerful method for solving the linear nonhomogeneous equations.

Theorem 13. Let . Then the function satisfies (69) and both boundary and transmission conditions (2)–(4), where is the Green function defined in (68).

Proof. Putting (68) in (70), we have By differentiating twice and by the definitions of solutions and , we obtain . So, the function defined in (70) is the solution of (69). And by calculations, we may prove that (71) satisfies both boundary and transmission conditions (2)–(4).

4. Eigenfunction Expansion for Green’s Function and the Modified Parseval Equality

In this section, we derive the eigenfunction expansion for Green’s function of the S-L problem (1)–(4) and establish the modified Parseval equality in the associated Hilbert space . Without loss of generality, we assume that is not an eigenvalue. Let .

Theorem 14. Let be the eigenvalue of the S-L problem (1)–(4) and let be the corresponding normalized eigenfunction. Then,

In order to prove Theorem 14, we need the following lemma.

Lemma 15. The S-L problem (1)–(4) is equivalent to

Proof. By Theorem 13, we know that satisfies and both boundary and transmission conditions (2)–(4). Equation (69) can be written in the form where . Similar to (74), the new form of (69) has a solution which satisfies both boundary and transmission conditions (2)–(4). If , then the corresponding homogeneous case is the S-L problem (1)–(4). Consequently, the S-L problem (1)–(4) is equivalent to

Proof of Theorem 14. Let be the eigenvalue of the S-L problem (1)–(4) and let be the corresponding normalized eigenfunction as in Theorem 11. Let . Then is continuous and symmetric. We assume that . Then, by the Fredholm integral equation, there are a number and a function in such that By Lemma 15, Putting in (78), we obtain In the following, we prove that and is an eigenfunction of the S-L problem (1)–(4). By (77) and (79) And we have by (77). This implies that is the eigenfunction of the S-L problem (1)–(4) by Lemma 15. So, by and the completeness of the eigenfunctions in Theorem 11, we know that . Consequently, . The proof is completed.

Lemma 16. Let be the set of all functions defined by where ,  . Then is dense in .

Proof. Let be any function in with , and . Since is dense in as in [23], for , there exists a function such that Similarly, for , there exists a function such that Then, for any and , there exists with such that Thus, is dense in .

In the following theorem, we prove the modified Parseval equality in the associated Hilbert space .

Theorem 17. Let . Then, the modified Parseval equality holds; that is, where and

Proof. Let be as in Lemma 16. At first, we prove that (86) holds for . Denote . Then by Theorems 13 and 14, Multiplying by and integrating the new equation, we have Then where . Thus, for ,
Next, we prove that (86) holds for . For any , there exists a sequence converging to in since is dense in . Firstly, we prove that and .
By the Cauchy-Schwartz inequality, . Hence, . Since for , Let , then inequality (92) becomes . Letting , we have Then by the Minkowski inequality